UNIVERSITY  OF  CALIFORNIA 


GIFT  OF 

Col.   Glen  F. 


PE.PRRTr«QE.1MT     DOCUMENT    NO.  ZO35 


THEORY  AND  DESIGN 

OF 

RECOIL  SYSTEMS 

AND 

GUN  CARRIAGES 


Prepared  in  the 
Office  of  the  Chief  of  Ordnance 


SEPTEMBER, 


REPRODUCTION  PUANT 
WA5H\NQTON  BARRRCKS  O.C. 


-rinf 
Lilrary 

U.F 


ORDNANCE   DEPARTMENT 
Document  No.  2035 

Off ic«  of  the  Chief  of  Ordnance 


WAR  DEPARTMENT, 

Washington,  October  1921. 

The  following  publication  entitled  "Theory  and 
Design  of  Recoil  Systems  and  Gun  Carriages"  is  pub- 
lished for  the  information  and  guidance  of  all  students 
of  the  Ordnance  training  schools,  and  other  similar 
educational  organizations.   The  contents  should  not  be 
republished  without  Authority. 

By  order  of  the  Secretary  of 


C.  C.  WILLIAMS, 

MAJOR  GENERAL,  CHIEF  OF  ORDNANCE. 


Jjlraiy 


FOREWORD 

This  edition  is  published  in  its  present  form  with 
lioeral  margins  and  spacing  so  that  corrections  or 
additions  may  be  freely  made.   In  a  document  of  this 
kind  it  is  almost  inevitable  that  ambiguities,  errors 
and  misstatenents  will  appear,  and  it  is  only  in  extended 
and  repeated  use  that  these  are  fully  exposed.   It  will 
therefore  be  appreciated  if  all  those  to  whom  this 
volume  comes  and  who  use  it  critically  will  forward 
criticisms,  corrections  or  necessary  additions  to  the 
Artillery  Division,  Ordnance  Office,  Washington,  D.  C., 
so  that  these  nay  be  incorporated  in  the  master  volume. 
After  such  changes  have  been  received  for  a  suitable 
period,  it  is  expected  to  have  the  text  printed  in 
usual  book  form. 


PREFACE 

Although  strictly  artillery  design  may 
be  considered  a  highly  specialized  branch  of 
machine  design,  there  are  so  many  features  that 
differentiate  this  work  from  ordinary  machine 
design,  it  has  been  felt  that  a  volume  covering 
the  specialized  points  is  of  fundamental 
importance  in  order  that  our  designing  engineers 
may  have  in  a  readily  accessible  from  reference 
data  covering  the  general  subject  and  in 
particular  those  features  of  modern  development 
not  now  covered  in  published  works.   Such  is  the 
purpose  of  this  volume. 

Artillery  design  may  be  subdivided  into  the 
design  of  cannon,  and  the  design  of  gun  carriages 
and  recoil  systems.   During  the  late  war  the  ex- 
tensive introduction  of  self  propelled  gun  mounts, 
such  as  caterpillar  vehicles,  has  introduced 
automotive  problems  in  the  design  of  these  types 
of  gun  mounts  in  addition  to  the  ordinary 
consideration  affecting  design  of  gun  mounts. 
Further  in  the  design  of  artillery  we  have  three 
important  aspects,  -  (l)  the  technical  and 
theoretical  considerations  of  a  design,  (2)  the 
fabrication,  standardization  and  production 
features,  and  (3)  the  service  and  field  require- 
ments to  be  fulfilled.   All  three  aspects  are 
equally  important  and  a  successful  design  results 
only  from  a  balanced  consideration  of  the  three. 

This  discussion  has  been  written  under  the 
auspices  of  Colonel  G.  F.  Jenks,  Chief  of  the 
Artillery  Division,  Ordnance  Department,  U.  S.  A. 
and  of  Colonel  J.  B.  Rose,  Chief  of  the  Mobile 
Gun  Carriage  Section  of  that  Division.   Effort 
has  been  made  to  arrange  systematically  in  a 
form  for  reference  tne  great  quantity  of 
engineering  data  in  the  files  of  the  office. 
In  order  to  develop  and  analyze  this  data,  it  has 


been  necessary  to  introduce  a  considerable  number 

of  original  discussions  and  deductions. 

The  work  is  an  attempt  to  cover  only  the 

*» 
technical  aspect  of  the  design  of  gun  carriages 

and  recoil  systems.   The  fabrication  and  field 
service  phases,  though  of  course  inherently 
coordinate  in.  a  design  are  subjects  of  such  com- 
plexity and  broadness  that  they  require  for  their 
full  appreciation  a  separate  treatment.  These 
aspects  have  therefore  necessarily  been  entirely 
onitted,  except  in  so  far  as  they  are  directly 
connected  with  the  technical  features  involved. 
Acknowledgement  and  thanks  are  especially 
due  to  Colonel  J.  B.  Rose,  who  has  proof  read  the 
complete  work  in  the  view  of  bringing  the  data  into 
conformity  with  the  practice  and  standards  of  the 
Ordnance  Department.   It  should  be  stated,  however, 
that  this  has  been  done  only  to  the  degree  which 
was  found  possible  without  destroying  original  con- 
clusions and  discussions  or  without  alteration  of 
the  system  of  nomenclature  used.   The  latter  is  in 
partial  but  not  complete  agreement  with  the  most 
general  practice.   Further  acknowledgement  and 
thanks  for  suggestions  on  the  various  parts  of  the 
work  are  due  to:  - 

Mr.  D.  A.  Gurney,  Ordnance  Engineer,  Mobile 
Gun  Carriage  Section, 
Artillery  Division. 
Prof.  B.  V.  Huntington,  Professor  of 

Mathematics  and  Mechanics, 
Harvard  University 

Professor  C.  E.  Fuller,   Professor  of  Ap- 
plied Theoretical 
Mechanics,  Massachusetts 
Institute  of  Technology. 
Professor  G.  Lanza,  Professor  Emeritus 

in  charge  of  Mechanical 
Engineering  Department, 
Massachusetts  Institute 
of  Technology. 


Acknowledgement  of  assistance  on  the  Computation 
work  is  due  to  Mr.  E.  V.  B.  Thomas,  Mr.  Kasargian 
and  Mr.  McVey  of  the  Artillery  Division,  also  to 
Messrs.  Murray  H.  Resni  Coff  and  0.  L.  Garver  for 
preparing  this  data  for  publication. 


RUPBN  BKSERGIAN, 
Formerly  Captain, Ordnance  Dept.U.S.A, 


Chapter  I 

Chapter  II 

Chapter  III 
Chapter  IV 

Chapter  V 


Chapter  VI 


Chapter  VII 


Chapter  VIII 
Chapter  IX 


Introduction  and  Elements  -  Types  of 
Cannon  and  Carriages  -  Classifi- 
cation of  Carriages  and  Recoil 
Systems. 

Dynamics  of  Interior  Ballistics  as  Af- 
fecting Recoil  Design  -  The  Para- 
bolic Trajectory. 

External  Reaction  on  Carriage  during 
Recoil  -  Stability  -  Jump. 


Internal  Reactions  throughout  a 
Carriage  during  Recoil  and 
Counter  Recoil. 


Gun 


Hydraulic  Principles  as  Applied  to 
Various  Systems  of  Recoil  and 
Counter  Recoil  -  General 

Theories  on  Orifices  and 
Flow  of  Oil. 

The  Dynamics  of  Recoil  and  Counter  Re- 
coil -  Differential  Equations  of 
Resistance,  Braking,  etc.  and 
of  Velocity  -  General 
"Formulas  for  Recoil 

and  Counter  Recoil, 

Classification  of  Recoil  Systems  - 

Derivation  of  General  Formulas  for 
Design  and  Computation  - 

General  Linitations,  etc. 

Hydro-pneumatic  Recoil  Systems. 

Hydro-pneumatic  Recoil  Systems 
(Continued) 


8 

Chapter  X     Railway  Gun  Carriages. 
Chapter  XI    Gun  Lift  Carriages. 
Chapter  XII   Double  Recoil  Systems. 

Chapter  XIII  Miscellaneous  Problems  — 

Discussions  of  Various  Types 
of  Carriages. 


CHAPTER  I 

TYPES  AND  PRINCIPAL  ELEMENTS  OF  CANNON  AND 
CARRIAGES. 

The  fundamental  principles  of  gun 
carriage  design  are  entirely  the  same 
as  those  of  engine  and  machine  design, 
and  it  is  the  object  of  this  volume 
merely  to  bring  out  the  specific  ap- 
plication of  these  principles  to  the  design  of  gun 
carriages . 

A  gun  carriage  is  a  machine  exercising  primarily 
the  following  functions: 

(1)  To  provide  a  fixed  firing  platform 
which  dissipates  the  energy  given  to 
the  recoiling  parts  in  reaction  to  the 
energy  imparted  to  the  projectile  and 
powder  gases. 

(2)  To  return  the  recoiling  parts  to  their 
initial  position  for  further  firing. 

(3)  To  provide  the  mechanism  for  elevat- 
ing the  gun  for  different  ranges  and 
angles  of  site,  and  for  traversing  the 
gun  for  changes  in  the  direction  of  fire. 

The  effect  of  allowing  the  gun  and  a  part  of  the 
carriage  to  recoil  is  to  reduce  many  times  the  stresses 
in  the  carriage  and  to  maintain  its  equilibrium.   A 
properly  designed  recoil  system  will  give  reactions 
consistent  with  the  strength  and  stability  of  the 
carriage,  and  a  smoothness  of  action  which  is  essent- 
ial for  long  service  and  accuracy.   The  success  of  one 
design  over  another  is  due  to  perfection  of  many  de- 
tails, which  insures  smooth  action  and  long  service 
and  to  a  judicious  compromise  between  many  opposing 
conditions  and  requirements. 

To  approach  the  study  of  carriage  design,  it  is 
necessary  to  know  the  elements  of  interior  ballistics 
and  the  characteristics  of  guns  for  meeting  different 

9 


10 


ballistic  conditions  in  so  far  as  these  affect  the 
form  of  carriage  and  determine  the  forces  acting 
upon  it.  These  subjects  will,  therefore,  be 
briefly  considered,  but  a  complete  discussion  must 
be  obtained  from  works  treating  them  specifically. 


From  one  view  point  a  cannon  may 
be  considered  as  a  tube  of  proper 
thickness  for  strength,  having  a 
chamber  in  the  rear  of  somewhat  larg- 
er diameter  which  contains  the  powder 
charge.   The  powder  charge  is  inserted  by  opening  a 
breech  block  in  the  rear  end  of  the  cannon.   This 
breech  block  necessarily  must  withstand  the  maximum 
powder  pressure  over  its  cross  section  and  a  power- 
ful locking  device  is  therefore  needed.   The  details 
of  this  mechanism  are  complicated,  but  need  not  be 
considered  in  carriage  design,  except  in  special 
cases  where  the  breech  mechanism  is  operated  during 
counter  recoil.   The  design  of  the  rifling  grooves 
and  capacity  of  powder  chamber  will  be  considered 
later. 

The  elements  of  a  gun  are  shown  in  figure  (1). 
"A"  is  the  powder  chamber,  "B"  the  rifled  portion 
of  the  bore,  "C"  the  breech  block,  "D"  the  gun  lug 
for  the  attachment  of  piston  rods,  which  restrain 
the  gun  in  recoil. 


—  * 

—  V- 

il  \  . 

-L 
fV 

^ 

The  caliber  of  a  gun  is  the  diameter  of  the  bore 
and  is  expressed  usually  in  millimeters  or  inches. 
Speaking  very  roughly,  small  guns  range  from  37  m/m 


11 


to  75  m/m,  and  are  suitable  for  mounting  on 
aeroplanes  or  for  use  with  infantry.  Light  field 
guns  range  from  75  m/m  to  105  m/m.   Ordinary  medium 
artillery  ranges  from  105  m/m  to  8  inches*   Heavy 
artillery  ranges  from  8  inches  upward.  The  above 
classification  refers  to  mobile  field  materiel  only. 

PQWTT7FRS  AND  flUNS      Carriages  are  designed  for  either 

howitzers  or  guns.   Howitzers  are 
for  high  angle  fire,  the  striking 
angle  being  generally  above  25  de- 
grees.  They  have  a  medium  or  low 

muzzle  velocity.   A  gun  is  designed  for  range  and,  there- 
fore, has  a  high  muzzle  velocity. 

The  angle  of  elevation  of  howitzers  is  usually 
between  20"*  and  70°and  the  muzzle  velocities  from  400 
to  1800  feet  per  second.   The  angle  of  elevation  of  a 
gun  is  usually  from  minus  5  degrees  to  plus  45  degrees 
with  muzzle  velocities  ranging  from  1700  to  3000  feet 
per  second.   The  angular  velocity  of  the  projectile  is 
also  considerably  higher  for  a  gun  than  for  a  howitzer. 
In  modern  practice  the  line  of  demarcation  between  guns, 
howitzers  and  mortars  has  become  somewhat  less  distinct, 
and  we  may  consider  all  of  them  as  cannon  which  decrease 
in  power  in  the  order  named  and  generally  for  use  at 
elevations  which  increase  ia  the  order  named. 

Against  aircraft,  firing  is  at  elevations  from 
0  to  80  degrees,  hut  the  muzzle  velocity  is  high;  hence, 
the  pieces  used  in  such  work  are  properly  classified 
as  "guns". 

The  traversing  limitations  of  a  gun  and  howitzer 
may  be  the  same  or  different  but  do  not  enter  in  the 
differentiation  between  a  gun  and  howitzer. 

The  muzzle  velocity  of  howitzers  being  lower  than 
that  of  guns,  it  is  possible  with  the  same  total  weight 
of  materiel  to  fire  a  much  heavier  projectile. 


12 


BECOIU BG_PABT-g     The  recoiling  parts  consist  of 
the  gun  together  with  the  various 

parts  attached  to  it  and  recoiling 
with  it.   We  have  two  methods  of 
arrangement  of  recoiling  parts: 

(1)  the  piston  rods  with  their  pistons 
attached  to  the  gun  lug  and  recoiling 
with  the  gua. 

(2)  the  pistons  and  their  rods  held 
stationary. 

So  far  as  the  recoil  mechanism  is  concerned  we  are 
only  concerned  with  the  relative  motion  between  the 
rods  and  pistons  and  their  cylinders. 

The  greater  part  of  our  guns  in  the  service 
translate  in  recoil  directly  along  the  axis  of  the 
bore,  others  as  on  certain  Barbette  mounts  and 
double  recoil  systems  have  a  translation  in  addition 
to  that  along  the  axis  of  the  bore.   Guns  on  Dis- 
appearing carriages  and  special  other  types  have 
rotation  in  addition  to  translation. 

In  ordinary  recoil  systems  the  center  of  gravity 
of  the  recoiling  parts  is  usually  located  slightly  below 
the  axis  of  the  bore.   This  insures  a  positive  jump 
(muzzle  up)  during  the  powder  pressure  period.   If 
the  center  of  gravity  of  the  recoiling  parts  is  great- 
ly below  the  axis  of  the  bore  considerable  stresses 
are  brought  upon  the  elevating  rack  and  pinion,  due 
to  the  fact  that  the  powder  pressure  causes  an  ex- 
cessive turning  effect  about  the  trunnions  the 
amount  depending  also  upon  the  location  of  these. 
For  this  reason  when  the  cylinders  recoil  with  the 
gun,  extra  weight  is  very  often  introduced  on  the 
top  of  the  gun.   This,  of  course,  raises  the  center 
of  gravity  of  the  recoiling  parts  nearer  the  axis 
of  the  bore. 

The  recoiling  parts  are  constrained  to  recoil 
parallel  with  the  axis  of  the  bore  by  gun  clips  en- 
gaging in  guides  in  a  fixed  cradle  or  by  the  gun 
itself  sliding  in  a  fixed  cylindrical  sleeve.   Due 
to  the  fact  that  the  braking  forces  developed  in  the 


13 


cylinders  are  usually  considerably  below  the  axis 
of  the  bore  during  recoil,  considerable  pinching 
action  takes  place  at  the  front  and  rear  clip  contact 
with  the  guides.   This  causes  somewhat  greater  friction 
than  would  be  obtained  by  mere  sliding  friction. 

The  clips  attached  to  the  recoiling  parts,  or 
rather  to  the  gun  itself,  which  in  turn  engage  in  the 
guides  of  the  cradle,  are  usually  either  continuous  or 
three  to  four  in  number.   In  order  to  maintain  a  con- 
stant friction  throughout  the  recoil,  clips  should 
be  evenly  spaced  along  the  gun  and  the  front  clip 
should  engage  in  the  guides  before  the  rear  clip  leaves 
the  guides.   When  the  gun  recoils  in  a  sleeve  or  cy- 
linder which  is  a  part  of  the  cradle,  it  is  conetiroes 
possible  to  distribute  the  various  pistons  and  cy- 
linders symmetrically  about  the  axis  of  the  bore. 
As  we  shall  see,  this  decreases  the  friction  during  the 
recoil  and  counter  recoil. 


Figure  (2)  shows  the  recoiling  mass  where  the 
pistons  and  their  rods  recoil  with  the  gun.   Below 
in  figure  (3)  is  shown  a  recoiling  mass  consisting 
of  the  cylinders  grouped  together  in  a  single  forging 
in  a  so-called  slide  or  sleigh,  and  rigidly  attached 
to  the  gun. 


14 


.  3 


nt£_CRADLE_        The  cradle  serves  as  a  constrain- 
ing member  for  the  sliding  of  tlie  gun 

to  the  rear  in  recoil  and  as  a  support 
for  elevating  the  gun.   The  cradle  and 
the  recoiling  parts  together  are  known 
as  the  tipping  parts  and  turn  about  horizontal  trun- 
nions fixed  to  the  cradle  and  resting  in  bearings  in 
the  top  carriage.  To  elevate  the  tipping  parts,  an 
elevating  arc  bolted  to  the  cradle  engages  in  a  pinion 
fixed  to  the  top  carriage,  or  vice  versa.  The  cradle 
and  therefore  the  tipping  parts  are  supported  in  the 
top  carriage  at  two  points: 

(1)  at  the  trunnions 
and 

(2)  at  the  tooth  contact  of  the 
elevating  arc  and  pinion. 

See  figure  (4). 


IS 


ng.  4- 


When  the  cylinders  do  not  recoil  they  are  in  turn 
an  integral  part  of  the  cradle,  and  therefore,  the 
recuperator  forgings  and  the  cradle  are  one  and  the 
same.   A  sleigh  may  or  may  not  he  interposed  between  the 
gun  and  cradle.   With  guns,  where  the  cylinders  recoil 
with  gun,  the  cradle  merely  serves  the  purpose  of  a 
constraining  guide  for  the  recoiling  parts  and  rigid- 
ly attached  to  it  are  the  piston  rods  and  their  pis- 
tons . 

TJPPIflG_FABT§      The  term"tipping  parts"  applies 

to  those  parts  of  a  carriage  which  move 
in  the  process  of  elevating  the  gun. 
In  order  to  rapidly  elevate  the  gun,  it 
is  considered  very  important  that  the 
tipping  parts  are  nicely  balanced  about  the  trunnions. 
Thus  the  center  of  gravity  of  the  tipping  parts  must 
be  located  at  the  trunnions.   As  the  height  of  the 
trunnions  and  axis  of  the  bore  are  governed  by 
stability  at  horizontal  elevation,  clearance  in 
traveling  and  accessibility  for  leading,  the  length 
of  recoil  at  maximum  elevation  becomes  limited.   If 
a  minimum  elevation  of  about  20  degrees  is  allowed  for 
a  howitzer,  we  might  raise  the  trunnions,  thereby  in- 


16 


crease  the  length  of  recoil,  and  thus  maintain 

stability.   When,  however,  a  gun  must  fire  at  high  ele- 
vation as  in  antiaircraft  materiel,  or  when  a  carriage 
serves  the  double  purpose  of  supporting  a  gun  or  a 
howitzer  at  high  elevations,  the  maximum  possible  re- 
coil at  maximum  elevation  becomes  greatly  limited. 
The  recoil  displacement  at  maximum  elevation  may 
be  increased  most  satisfactorily  by  placing  the 
trunnions  to  the  rear  and  introducing  a  balancing 
gear  for  balancing  tbe  tipping  parts  about  the 
trunnions. 

The  balancing  gear  usually  consists  of  an  os- 
cillating spring  or  pneumatic  cylinder,  the  trunnions 
of  which  rest  in  bearings  in  the  top  carriage,  the 
end  of  the  piston  rod  being  attached  to  the  cradle. 
Since  it  is  difficult  to  obtain  perfect  balance  by 
this  method  throughout  the  elevation,  the  maximum 
unbalanced  moment  in  the  process  of  elevation  should 
be  considered  in  the  design  of  the  elevating  gear 
mechanism.   A  method  by  which  exact  balance  can  be 
maintained  throughout  the  elevation  is  obtained  by 
use  of  a  cam  and  chain  connecting  tbe  cradle  with 
the  spring  or  pneumatic  cylinder.   In  this  case  the 
cam  is  fixed  to  the  cradle  and  the  spring  cylinder  to 
the  top  carriage.   However,  due  to  the  variation  in 
trunnion  friction  and  other  similar  factors  the 
former  method  is  probably  better  since  a  very  close 
approximation  in  balance  throughout  the  elevation  can 
be  obtained. 

The  reaction  on  the  elevating  arc  and  the 
trunnion  reaction  are  modified  by  the  introduction 
of  tbe  balancing  gear,  though  ordinarily  where  the 
weight  of  tipping  parts  is  relatively  small  as  com- 
pared with  the  recoil  reaction  the  effect  of  the 
balancing  gear  on  the  reactions  may  be  neglected. 

When  it  is  desired  to  use  an  independent  line 
of  sight,  a  rocker  is  introduced  between  the 
elevating  pinion  and  cradle.  The  rocker,  when  moving, 
is  a  part  of  the  tipping  parts.   In  the  process  of 


17 


elevating  the  gun  an  elevating  pinion  rotates 

the  rocker  about  the  trunnions  until  the  proper 
line  of  sight  is  obtained;  the  cradle  is  then 
brought  into  its  proper  position  by  gearing  con- 
necting the  rocker  and  cradle. 

TOP  CARRIAGE      The   top  carriage  serves  as  an. 
intermediary  piece  connecting  the 
tipping  parts  with  the  bottom  car- 
riage, or  in  semi-fixed  mounts,  with 
the  bottom  platform.   The  top  car- 
riage is  supported  at  its  bottom  by  a  vertical 
pintle  block  and  circular  traversing  clips.   At 
the  top  it  supports  the  tipping  parts  on  its  trun- 
nion bearings  and  elevating  pinion  bearing.   The 
top  carriage  -together  with  the  tipping  parts  are 
known  as  the  traversing  parts.   To  traverse  the  gun, 
the  top  carriage  with  the  tipping  parts  are  rotated 
in  a  horizontal  plane  about  the  pintle  block  by  a 
circular  traversing  reck  and  pinion  or  worm  gear. 
In  certain  types  of  field  artillery  the  top 
carriage  is  an  integral  part  of  the  trail,  in  which 
case  traversing  is  obtained  with  respect  to  the 
wheels  and  axle  by  moving  the  trail  along  the 
axle  and  about  the  spade  point  as  a  pivot.   Traverse 
by  this  method  is  naturally  very  limited  as  com- 
pared to  traverse  with  a  rotating  top  carriage.   All 
stationary  mounts  or  field  platform  mounts  have  a 
separate  top  carriage  which  serves  this  specific  function 
of  traversing  about  the  vertical  pintle  support.   In  very 
large  carriages  the  top  carriage  is  supported  by  a 
circular  ring  of  horizontal  rollers,  the  pintle 
bearing  merely  serving  as  a  constraining  pivot.   In 
certain  types  where  the  bottom  carriage  itself  is 
traversed,  the  top  carriage  is  used  for  translation 
only.   It  is  then  supported  on  rollers  moving 
along  an  inclined  or  horizontal  plane  and  the  braking 
is  affected  by  a  recoil  cylinder  in  the  top  carriage  which 


18 


connects  the  top  carriage  with  the  bottom  carriage 
through  the  piston  rods. 


Fig.  5 

Top  carriages  way  be  roughly  classified  into: 

(1)  the  ordinary  type  of  side  frames 
connected  at  the  front  or  rear  by 
cross  beams  or  transoms  which  contain 
the  pivot  bearing. 

(2)  pivot  yoke  type  used  on  small 
mobile  mounts 


and 


(3)     trail  carriages. 

The  ordinary  side  frame  type  of  top  carriage  is 
extensively  used  on  the  stationary  mounts  on  mobile 
platform  mounts  and  even  on  trail  supported  carriages. 
The  pivot  yoke  type  is  especially  useful  when  split  trails 
are  introduced,  since  it  supports  the  equalizer  bar  for 
balancing  the  distribution  of  the  load  between  the 
two  trails. 


19 


TRAIL  AND  SPADE     With  mobile  field  artillery 

it  is  customary  to  use  a  trail  and 
spade  for  the  double  purpose  of 
preventing  a  backward  motion  of  the 
carriage  on  firing  of  tbe  gun,  and 
of  giving  sufficient  stability  to  the  carriage  in 
order  that  the  wheels  may  not  leave  the  ground. 
We  have  two  classifications  of  trails,  -  (1)  the 
single  or  box  trail,  (2)  the  split  trail.   With  a 
single  trail  it  is  necessary  to  have  a  large  U- 
shaped  aperture  or  fork  arrangement  at  the  forward 
end  in  order  to  elevate,  load  and  traverse  the  gun 
without  interference.  When  split  trails  are  used 
we  have  two  separate  single  trails  which  may  turn 
at  the  wheel  ends  about  the  axle.   It  is  customary 
with  the  split  trails  to  introduce  an  equalizing 

mechanism  which  connects  the  two  trails  and 
distributes  the  load  between  the  trails  on 
firing. 

The  spade  and  float  support  tbe  trail  and  are 
designed  to  take  up  the  horizontal  and  vertical  re- 
actions at  the  rear  end.   In  tbe  design  of  the 
spade  and  floats  it  is  important  that  tbe  unit  bear- 
ing pressure  be  held  to  a  low  value.   This  should 
not  be  more  than  about  30  Ib.  per  sq.  in.  for  the 
float  and  40  Ib.  per  sq.  in.  for  the  spade. 

For  wide  traverse  of  the  gun  it  is  necessary 
to  lift  the  spade  from  the  ground  and  turn  the 
carriage  to  the  desired  line  of  fire.  For  this 
reason  the  static  load  on  the  spades  should  not  ex- 
ceed about  100  Ibs.  for  light  carriages.   Thus  in 
a  preliminary  lay-out  of  tbe  carriage,  it  is 
necessary  to  locate  the  center  of  gravity  of  tbe 
total  system  in  battery  very  close  to  the  axle  in 
order  that  the  static  load  under  the  float  does  not 
exceed  the  desired  amount.   This  inherently  makes 
the  counter  recoil  stability  in  battery  very  small 
especially  at  horizontal  recoil  and  requires  con- 
siderable care  in  the  design  of  a  counter  recoil 


20 


system. 

At  horizontal  elevation  the  carriage  is  usually 
designed  with  a  very  small  margin  of  stability. 
Therefore,  in  firing  the  vertical  load  on  the  float 
practically  equals  the  weight  of  the  total  system. 
The  bending  moment  in  the  trail  gradually  increases 
from  the  spade  toward  the  wheel  axle.   We  have  max- 
imum bending  moment  at  the  attachment  of  the  trail 
to  the  top  carriage  or  wheel  axle. 

PLATFORM  MOUNTS.    With  fixed  mounts  and  heavier 
types  of  field  artillery  it  is 
customary  to  support  the  travers- 
ing parts  on  a  platform,  that  is, 
the  top  carriage  rests  upon  a 

platform  which  serves  as  a  bottom  carriage.   When 
a  platform  bottom  carriage  is  used,  it  must  be 
either  bolted  to  a  concrete  foundation  as  in  fixed 
mounts  or  else  it  must  have  a  vertical  projection 
similar  to  a  spade  on  a  field  carriage  to  take  up 
the  horizontal  reaction  in  firing.   Further,  the 
bearing  surfaces  of  this  platform  must  be  suf- 
ficient to  prevent  overturning  of  the  carriage 
firing  at  low  angles  of  elevation  or  change  in 
level  in  firing  at  any  elevation.   That  is,  the 
center  of  pressure  of  the  reaction  of  the  earth 
must  be  within  the  middle  third  of  the  length  or 
diameter  of  the  platform  in  the  line  of  fire. 
Since  platform  mounts  vary  considerably  in  con- 
struction of  detail  no  attempt  will  be  made  to 
catalogue  the  various  types  used. 

With  fixed  mounts  the  bottom  carriage  or  plat- 
form is  usually  secured  to  a  concrete  foundation 
by  a  distribution  of  bolts  along  a  circular  flange; 
and  since  with  fixed  mounts  all  round  traverse  is 
possible,  each  bolt  should  be  designed  for  maximum 
tension. 


21 


DC 


22 


CATERPILLAR  MOUNTS.       To  increase  mobility 

during  the  World  War,  cater- 
pillar mounts  were  developed 
extensively.   A  caterpillar 
mount  consists  of  an  ordinary 

gun  mount  including  the  tipping  parts  and  top  car- 
riage mounted  on  a  bottom  carriage  which  fits  with- 
in the  frame  of  the  caterpillar.   The  caterpillar 
is  propelled  by  its  own  engine,  and  traverse  can  be 
readily  made  by  keeping  one  of  the  caterpillar 
tracks  stationary  and  moving  the  other.   For  more 
delicate  traversing  the  top  carriage  is  provided 
with  limited  traverse  about  the  bottom  carriage. 
The  essential  features  of  the  caterpillar  proper 
are: 

(1)     The  frame  which  supports  the 
bottom  carriage  and  the  principal 
bearings  for  the  driving  mechanism. 
The  caterpillar  frame  in  turn  is 
generally  supported  on  a  series  of 
roller  trucks  which  travel  on  the 
caterpillar  tracks. 

Between  the  roller  trucks  and  caterpillar 
frame,  spring  supports  are  usually  provided,  and 
the  roller  trucks  are  built  to  have  more  or  less 
up  and  down  movement  at  their  ends  to  conform  with 
the  contour  of  the  ground. 

The  frame  may  be  either  a  casting  or  built  up 
of  structural  steel.   The  structural  steel  frame  is 
perhaps  lighter  but  more  subject  to  objectional  de- 
flections .   .,» 

The  reactions  on  the  frame  consist  of  the 
various  spring  supports  from  the  supporting  roller 
trucks,  the  reactions  of  the  bearings  of  the 
running  gear  and  the  reactions  of  the  gun  mount 
transmitted  by  the  bottom  carriage  to  the  frame  on 
firing. 


23 


The  frame  of  a  caterpillar  is  subjected 
to  a  complicated  system  of  stresses.  Due  to  various 
possible  loading  conditions  during  traveling  such 
as  the  entire  weight  of  the  caterpillar  being 
carried  in  the  center  or  else  at  the  ends,  we  have 
different  types  of  loading  reactions.  Further  a 
wrenching  action  with  corresponding  large  transverse, 
stresses  are  induced  by  the  supporting  reactions 
being  on  either  side  at  the  further  extremities  of 
the  track.   This  requires  considerable  lateral 
bracing.   In  fact  outside  of  fabrication  and  con- 
struction considerations,  the  design  of  the  cater- 
pillar frame  should  be  based  on  a  careful  analysis 
of  the  various  types  of  supporting  reaction  com- 
binations that  may  take  place  in  the  traveling  of 

the  caterpillar.   It  will  be  usually  found  that  the 
traveling  stresses  are  somewhat  greater  than  the 
firing  stresses  and  are  often  of  an  opposite 
character. 

The  driving  mechanism  of  the  caterpillar  con- 
sists of  two  tracks  each  consisting  of  a  continuous 
track  or  belt  of  linked  shoes.   The  caterpillar 
track  is  driven,  by  sprockets  usually  at  the  rear 
end.   The  drive  shaft  contains  at  one  end  the  track 
sprocket,  and  at  the  other  end  the  drive  sprocket 
gear,  which  meshes  by  a  suitable  gearing  to  a 
clutch,  the  system  of  gearing  &nd  clutch  being 
symmetrically  the  same  for  either  track.   The 
clutches  are  driven  by  bevel  gears  or  other 
forms  of  reduction  gearing  through  a  gear  box, 
and  sometimes  a  master  clutch,  to  the  engine 
crank  shaft.   The  traction  gearing  is  straight 
forward  and  is  very  similar  to  other  types  of 
drive  gear  transmission.   Mechanical  steering 
is  obtained  by  operating  either  the  right  or 
left  track,  holding  it  stationary  or  sometimes 
reversing  the  motion  and  running  the  track 
backward. 


24 


Electric  drive  caterpillar  mounts  are  in  two 
(2)  units  and  possess  certain  advantages';  first, 
the  transmission  can  be  greatly  reduced  in  either 
unit  by  the  use  of  compact  motors  a_nd  gearing'; 
second,  the  units  can  be  made  similar  and  the 
mobility  thereby  increased;  third,  a  better  design  cf 
gun  mount  is  possible  due  to  less  limitations  on 
clearance  and  other  corresponding  factors.   The 
electric  drive  consists  of  the  gun  mount  unit  and 
the  power  plant  unit.   The  power  plant  unit  sup- 
plies power  for  driving  itself,  as  well  as  the 
gun  mount;   fourth,  the  caterpillar  is  braked  in 
traveling  by  suitable  band  brakes  in  the  trans- 
mission.  When,  however,  the  gun  is  fired,  it  is 
necessary  to  brake  the  caterpillar  from  running 
back.   The  braking  and  torque  being  usually  in  an 
opposite  direction  and  necessarily  of  a  large 
value  as  compared  with  the  traveling  braking1;  it 
is  usually  customary  to  introduce  a  band  brake  on 
the  final  drive  shaft  and  thus  eliminate  the 
stresses  in  the  transmission  during  firing.  The 
braking  should  be  designed  to  produce  a  traction 
reaction  equal  to  approximately  80  percent  of  the 
total  caterpillar.  Fifth,  in  a  design  of 
caterpillar  mounts,  stability  is  of  prime  im- 
portance due  to  the  limited  wheel  base  and 
necessity  of  maintaining  as  light  a  mount  as 
possible.   Stability  may  be  increased  by  the  use 
of  outriggers  attached  to  the  caterpillar  body. 
To  decrease  the  overturning  reaction  of  the  recoil 
on  firing  and  thus  increase  the  stability,  double 
recoil  systems  have  been  successfully  introduced  on 
larger  caterpillar  guns.   A  double  recoil  system  consists 
of  an  ordinary  recoil  system  between  the  gun  and 
cradle  of  the  top  carriage  and  a  lower  recoil 
system  between  the  top  carriage  and  frame.   The 
top  carriage  is  designed  to  roll  up  an  inclined  plane 
of  sufficient  elevation  to  bring  the  recoiling 
masses  into  battery  and  the  cate-rpillar  lies  in  a 


r 

k 

tf 


26 


horizontal  plane.   This  elevation  is  usually  at 
from  6  to  7  degrees.   By  the  use  of  double  re- 
coil systems  fhe  stability  is  greatly  enhanced,  since 
the  inertia  resistance  of  the  top  carriage  creates 
a  stabilizing  moment  which  is  added  to  the 
inertia  resistance  of  the  upper  recoiling  parts. 
In  the  design  of  the  double  recoil  system  cater- 
pillar mount,  it  is  highly  desirable  that  the  top 
carriage  recoil  as  far  as  possible  up  the  inclined 
plane.   Due  to  less  limitations  and  clearance,  an 
electric  drive  of  the  two  supporting  units  offers 
a  very  suitable  gun  mount  and  a  long  recoil  of  the 
lower  recoil  system  is  usually  possible. 


Fig.  6 


RAILWAY  MOUNTS      Railway  mounts  developed 

during  the  late  war  consist  of 
three  (3)  systems:  (1)  these 
where  the  car  mounted  on  suitable 
tracks,  rolls  back  on  firing;(2). 
those  sliding  back  on  a  special  track  the  tracks 
being  disengaged,  (3)  platform  or  stationary 
railway  mounts  with  suitable  outriggers,  the 


27 


trucks  being  entirely  disengaged.   In  types  (1) 

and  (2)  a  very  limited  traverse  is  possible, 
whereas,  in  type  (3)  considerable  amount  of  traverse 
is  possible. 

Railway  mounts  of  type  (1),  rest  upon  suitable 
girders,  supported  by  the  trucks  at  either  end.   The 
girder  must  be  designed  to  carry  the  maximum  firing 
load  stresses  at  maximum  elevation,  as  well  as 
stresses  due  to  the  dead  load  weights.   The  trucks 
take  the  supporting  reactions  from  the  girders  of 
the  dead  weight  load  as  well  as  fhe  firing  load  at 
maximum  elevation.  Great  care  is  needed  in  dis- 
tributing the  loading  from  the  various  axles  by 
properly  formed  truck  equalizers. 

In  type  (2)  a  special  built-up  track  is 
necessary,  the  trucks  being  disengaged  merely 
carrying  the  dead  weight  of  the  mount.   The  mount 
is  designed  to  have  a  considerable  bearing  surface, 
and  thereby  the  bearing  pressures  are  greatly  reduced 

In  sliding  railway  types,  recoil  systems  have 
in  certain  types  been  completely  eliminated,  the 
recoil  being  merely  resisted  by  the  friction  of 
the  track.   Due,  however,  to  the  enormous  stresses 
due  to  high  caliber  guns  at  maximum  elevation, 
recoil  systems  should  always  be  introduced. 

With  stationary  or  platform  mounts  the 
question  of  stabilizers  of  corresponding  outriggers 
become  a  fundamental  feature  in  this  type  of  design. 
Platform  railway  mounts  have  similar  characteristics 
as  ordinary  field  platform  mounts  in  mobile 
artillery. 


28 


rig.  9 


Fig.  10 


CHAPTER   II 

DYNAMICS  OF  INTERIOR  BALLISTICS  AS  AFFECTING  RECOIL 
DESIGN. 

The  object  of  interior  ballistics  is  partly  to 
derive  expressions  for  the  acceleration  and  velocity 
of  the  projectile  during  the  travel  in  the  bore,  and 
the  corresponding  pressures  on  the  base  of  the  shell 
and  breech  in  terms  of  tne  powder  loading,  the  form 
of  powder  grain,  the  initial  volume  of  powder  chamber 
in  the  gun,  and  other  variables  upon  which  the 
velocity  and  pressure  depend.   In  the  design  of  the 
recoil  mechanism  as  well  as  the  carriage  for  its 
maximum  stresses,  it  is  very  important  to  know  the 
accelerations,  velocities,  and  pressures  in  the  gun 
to  a  considerable  degree  of  accuracy  throughout  the 
time  the  powder  gases  act. 

In  the  study  of  interior  ballistics,  it  is  con- 
venient to  divide  the  powder  pressure  interval  into 
two  periods: 

(1)  The  interior  period  while  the  shot 
travels  up  the  bore  to  the  muzzle. 

(2)  The  after  effect  period  while  the 
powder  gases  expand  after'  the  shot  has 
left  the  muzzle. 

During  the  interior  period,  we  have  considerable 
combustion  of  the  charge  and  corresponding  gas  evolved 
in  the  powder  chamber  before  the  shot  has  left  its 
initial  position  in  the  breech  end  of  the  bore,  the 
temperature  rising  and  the  pressure  reaching  a 
value  sufficient  to  force  the  projectile  into  the 
rifling  groove  and  to  overcome  initial  frictions, 
usually  a  considerable  fraction  of  the  powder  pres- 
sure obtained.   The  projectile  then  moves  up  the 
bore  followed  by  further  combustion  and  expansion 
of  the  gases  evolved  from  the  combustion  of  the 
powder.   The  combustion  exceeds  the  expansion  up 


29 


30 


to  the  tine  of  maximum  powder  pressure  which  is 
reached  after  a  travel  up  the  bore  roughly  from 
1/6  to  1/3  the  length  of  the  bore  depending  greatly 
on  the  type  of  cannon,  charge,  etc. 

The  energy  of  combustion  is  expended: 

(a)  In  Kinetic  Energy  of  translation 
?  of  the  projectile. 

(b)  In  Kinetic  Energy  of  translation 
of  the  recoiling  mass  (assuming  the 
recoiling  mass  free). 

(c)  In  the  Kinetic  Energy  of  the 
charge  itself. 

(d)  In  the  work  on  the  rifling  and  in 
friction. 

(e)  In  the  angular  energy  given  to  the 
projectile. 

(f)  In  dissipated  heat. 

The  last  three  are  very  small  as  compared  with 
(a),  (b)  and  (c).  Further  (b)  and  (c)  are  small  as 
compared  with  (a). 

"Ingalls"  states  that  about  83*  of  the  total 
energy  of  the  work  of  expansion  goes  into  the 
Kinetic  Energy  of  translation  of  the  shot,  the  re- 
mainder 17*  going  into  the  forms  b,  c,  d,  e  and  f. 

The  rate  of  combustion  depends  upon  the  forn 
and  size  of  the  grain,  it  being  an  observed  fact 
that  powder  burns  in  layers  always  parallel  to  the 
initial  surface.   Further  the  rate  of  combustion  is 
a  function  of  the  actual  pressure  generated,  vary- 
ing as  some  power  of  the  pressure.   The  value  used 
for  this  exponent  is  one  of  the  most  tentative 
features  in  the  whole  subject  of  interior  ballistics. 


DYNAMIC  RELATION-     Let  IT  =  the  mass  of  the  pro- 
jectile. 
*  =  the  weight  of  the 


SHIPS  IN  INTERIOR 


BALLISTICS. 


projecti le . 
m  =  the  mass  of  the 


charge 


31 


it  =  the  weight  of  the  charge. 

mr=  the  mass  of  the  recoiling  parts. 

wr=  the  weight  of  the  recoiling  parts. 

u  =  the  travel  up  the  bore. 

x  =  the  absolute  displacement  of  the  shot  in  the 

bore . 
X  =  the  corresponding  displacement  of  the  recoiling 

parts . 

v  =  the  absolute  velocity  of  the  shot  in  the  bore. 
vo=  the  muzzle  velocity  of*  the  shot. 

V  =  the  free  velocity  of  the  recoiling  parts 

(absolute ). 
Pxj=  the  total  pressure  on  the  breech. 

P  =  the  total  pressure  on  the  base  of  the  shot. 
Pk=  the  intensity  of  pressure  on  the  breech 

(Ibs.  per  sq .  in. ) . 
p  =  the  intensity  of  pressure  on.  the  base  of  the 

shot  (sq.  in.), 
f  =  the  component  of  the  rifling  reaction  parallel 

to  the  axis  of  the  bore. 


Then, 

dv 
P  -  f  =  m  -—   ,  for  the  motion  of  the 

d  t, 

projectile 

dV 
and   P^  -  f  =  mr  —  ,  for  the  motion  of  the 

recoiling  mass  in  free 
recoil  (2) 

and  further  assuming  the  charge  to  expand  in 
parallel  laminae  with  the  successive  laminae 
having  velocities  as  a  linear  function  of  the 
end  velocities,  we  have, 

p.  _  p  =  5  /dv      dV  . 

D         9  V -    —  )  f  n\ 

2  dt      dt  (3> 


where 


dv   dV 

dt   dt    =  the  mean  acceleration 


32 


of  the  powder. 

Combining  (l),  (2)  and  (3) 

I  ,  dV        ffi  x  dv 

<*<•  *  — >  at  =  (n  +  T ?  at       (4) 

Integrating  successively, 

(m  *  JL)  V  ••<•  +  4-)v  (5) 


(mr  *  -1-)  x  -  (»  +  ~)x  (6) 

c  £ 

The  absolute  displacement  of  the  shot  in  the 
bore  is  connected  with  the  travel  (u)  up  the  bore 
by  the  following  relation: 

x  =  u  -  X 

since  the  positive  value  of  X  is  assumed  opposite 
to  x. 

Substituting  in  (6),  we  have, 

i  . 

(m  +  — - )  u 
X  -  2_ (7) 

mr+  m  •*•  m 

which  gives  the  relation  of  free  recoil  to  the 
travel  of  the  shot  up  the  bore. 

Obviously  (5),  (6)  and  (7)  may  be  written 
immediately  from  the  principle  of  "linear 
momentum"  (that  is,  the  total  momentum  of  the 
system  remains  constant  unless  acted  on  by  external 
forces)  and  the  principle  that  tha  center  of 
gravity  remains  fixed  unless  acted  upon  by  external 
forces.   In  free  recoil  the  exterior  forces  are 
nil. 

The  pressure  on  the  breech  exceeds  that  on  the 
shot  by  the  inertia  resistance  offered  by  the 
of  the  powder  gases, 


33 


>nrdV 
mdv 


Neglecting     ^       as   small  compared  with  mr, 
2 


pb   -  f          (m 


P  -  f 
hence 


Since  the  rifling  reaction  expecially  during 
the  movement  of  the  shot  up  the  bore  is  roughly  2 
per  cent  or  less  of  the  value  of  p,  we  may  entire 

ly  neglect  the  term  f  . 

' 


in  the  above  expression,  which  simplifies  to 

ft 
m  +  — 

Pb  =  -  2P  (8) 

• 

From  a  series  of  experiments  conducted  by 
the  United  States  Navy  the  value 

E 
m  +  -  - 

2m 
-  =   1.12  a  constant,  approx. 

• 

hence 

pb-  =  1.12  F  approx.  (9) 

It  is  to  be  noted  that  the  acceleration  of 
the  powder  is  very  likely  somewhat  different 

from  the  assumption  upon  which  (8)  was  de- 
rived, but  nevertheless  equations  (8)  and  (9)  give 
a  good  approximation  of  the  increase  of  breech 
pressure  over  that  at  the  base  of  the  projectile. 
During  the  "forcing  in  of  the  rifling"  before 
the  commencement  of  motion  of  the  shot,  obviously 
Pb  =  P- 


34 


According  to  the  previous  assumptions  the  pres 
sure  varies  progressively,  decreasing  from  its 
maximum  value  at  the  breech  block   to  a  slightly 
smaller  value  at  the  base  of  the  projectile. 

Therefore,  if  we  let  p^  be  the  average  or  mean 
instantaneous  pressure  or  rather  the  pressure  in 
the  -powder  chamber  and  bore,  we  have,    p  +  p 

•  P  =  -— 

*»     2 

In  terms  of  the  total  pressure  at  the  base  of  the 

projectile, 

m  dV     dv  m 

p  ,  .        7T  .  ,21   -3T,  dv 


m        *          •   2^ 


dt 


but  dv 

hence 

-•_ =  p=p(i  +  ^_)  =  p(1+  _L_)   (10) 

»  4  in  Aw 

or  in  terras  of  the  total  breech  pressure 

dV    "rar  dV 
mr  dt  +  ^»_  ~tt 

Pm  «  5 ~  (1 


m 
"  *   2 


(11) 


ID  *   1 

2 


35 


EQUIVALENT The  riflil1g  grooves  in 

MASS  OF  the  gun  come  in  contact  with 

PROJECTILE        the  copper  rifling  band  on  the 
projectile  and  angular  motion 
is  transmitted  to  the  projectile 

in  addition  to  the  translatory  motion.   The  object 
of  the  angular  motion  is  to  give  the  projectile  a 
gyroscopic  effect  maintaining,  with  a  combination 
of  the  air  reaction,  the  axis  of  the  projectile 
parallel  to  the  tangent  of  the  trajectory  and 
further  making  an  oblong  projectile  possible  with 
greater  ballistic  efficiency. 

Let 
P  =  the  reaction  of  the  powder  on  the  base  of  the 

shell. 

m  =  the  mass  of  the  projectile, 
f  =  the  total  rifling  reaction  normal  to  the  rifling 

groove, 
uf  =  the  friction  component  of  the  rifling  reaction 

tangent  to  the  rifling  groove. 
6  =  the  angle  of  pitch  of  the  rifling,  (i.  e.  the 

angle  the  rifling  makes  with  the  axis  of  the 

bore ). 

p  =  the  pitch  of  the  rifling 
d  =  the  diameter  of  the  bore 
k  =  the  radius  of  gyration  of  the  projectile.. 
x  =  the  displacement  of  the  projectile  up  the  bore 

from  its  initial  position. 
£5  =  the  corresponding  angular  displacement  twist  of 

the  projectile. 

Then   we   have, 

P  -  f(sin  e    4   u  cos  9    )   =  m  4!*- 

dt  \i£>) 


f(cos  e   -  u  sin  9    )  |     =   mk'  £f       (^} 


36 


Further  since,  the  number  of  complete  turns 
or  revolutions  of  the  projectile  in  its  linear 
displacement  x  or  its  angular  displacement  ft, 

is 

JL  or  JL 
p      2* 

i»e  have 

da#  =   2*   dax        (14) 

ar*"   ~p~  "31"*"" 

In  terms  of  the  angle  of  pitch  of  the 
rifling, 

-  16  =  x  tan  e     or  n   tan  e 

=  "" 


hence 

2        d*x 

T  tan  6  dF~     (is) 

Substituting  (14)  or  (15)  in  equation  (13) 
we  have 

*  dax  d*x 

mk*   4  "P"  dT»         mkMtan 

---  —  --------  =  -  - 


(cos  9  -  u  sin  e)d    (cos  9  -  u  sin  e)da 

(16  ) 

which  shows  the  reaction  f  is  always  proportional 
to  the  linear  acceleration  of  the  projectile. 
Therefore,  the  friction  uf  ,  is  also  proportional 
to  the  linear  acceleration. 

Substituting  (16)  in  (12),  we  have 


or   in   terms  of    the   rifling   angle, 

.sin  e    *   u  cos      e      \    4k*tan@.  d*x 

P  -    I   1  ^cos  6  -   u  51B  b       >  ~li  -  J    ffi     ip 


37 


which  shows  that  the  powder  reaction  P  is  also 
directly  proportional  to  the  linear  acceleration 
of  the  projectile.   Evidently  the  equivalent  mass 
of  the  projectile,  is 


.sin    e    «•   i 

i  cos  6    < 

cos  e  -  i 

i   sin   6  j 

.sin   Q   +   i 

i  cos   6    > 

^cos  e  -  ' 

j   sin   e    ; 

dp 


4  tan  e  ^a  -,      0\ 
" 


Hence  the  rifling  reaction  and  friction  due  to 
rifling  are  directly  proportional  to  the  powder  re- 
action, that  is  the  pressure  on  the  rifling  grooves 
always  varies  at  any  instant  directly  with  the  powder 
reaction. 

fbus  we  have  the  relationship  that  rifling. 
frlc.tj.on  behaves  exactly  like,  an  additional  mass: 
that  jLaf  it  has  an  inertia^ef  f  ect  since  it  is_prft- 
portional  to  the  aeeeleratipn. 

The  true  equivalent  mass  due  to  the  linear  and 
angular  inertia  of  the  projectile  alone,  can  be  ob- 
tained by  assuming  the  rifling  friction  zero,  (  i.e., 
putting  u  =  o) 

4  n8k*v 
n'  •  (1  —  -  -  )  m 


(i  "  ::"  °  )  .    (20) 


The  true  equivalent  mass  may  be  readily  checked  by 

a  consideration  of  the  total  energy  of  the  projectile,. 

that  is, 


\  m'v2  «  i  mva  +  y  In* 


2nv     2v  tan 

and  w  •  -  - 


38 


and  I  =  mka   where  k  =  radius  of  gyration  about 

its  longitudinal  axis. 

hence  .  »  .  * 

4n  k  .      ,,   4k*  tan*  9  N 

B1  -  (  1  +  = — )  m  =  (1  +  — )  m 

D*  d* 


EQUIVALENT  MASS       For  a  differential  layer  of 
OF  the  powder  charge  at  the  base  of 

POWDER  CHARGE      the  projectile,  its  velocity 
evidently  is  equal  to  that  of 
the  projectile  while  for  a  dif- 
ferential layer  at  the  breech,  the  velocity  is  equal 
to  that  of  the  gun.  For  intermediate  layers,  we  must 
assume  some  law  of  variation  of  velocities,  between 
the  two  end  limits.   For  simplicity  and  probably 
a  fairly  close  approximation,  we  will  assume  for  the 
various  laminae,  a  linear  variation  of  velocity  be- 
tween the  end  limits.   Further  since  the  velocity 
of  the  gun  is  small  as  compared  with  that  of  the 
projectile,  in  virtue  of  the  approximation  of  the 
whole  analysis,  we  are  entirely  justified  in  assum- 
ing the  recoil  velocity  entirely  negligible. 
If, 

Velocity  of  projectile  *  v  (ft.  sec.) 

Distance  between  breech 

and  base  of  projectile  =  x  (ft.) 

Velocity  of  any  inter  - 

mediate  lamina  =  v1  (ft/sec.) 

Distance  from  breech  to 

lamina =  x1  (ft.) 

Then 

v1  =  —  v   (ft. sec.) 
u 

If  we  assume  the  density  of  the  powder  is 
uniform  through  the  distance  x,  so  that  the 
weight  of  the  lanina  is  x  dx,  then  the  kinetic 


39 


energy  of   the   lanina   is 

W     v  '  W       V  2  o 

V  1     I  "          *  I  £•    *     • 

TT  o dx   or  — .   -7T    *    *X 

2  g         x»   2  g 

and  the  Kinetic  energy  for  the  total  charge 
becomes, 

K.  E.  of  w  -  f.   _Ii  rx  x'2  dx1 
x»   2g  ^o 


'  J  (i.  )  T-         (21) 


That  is,  the  equivalent  mass,  when  dealing 

with  the  energy  equation,  is  1/3  the  mass  of  the 
total  charge. 

It  is  important  to  note  that  when  dealing  with 
momentum,  the  momentum  for  the  total  charge  becomes, 
on  the  same  assumption 

w   u  x1         w 
JL  /   JLvdx'=.JLv      (22) 
gx  o  x         2g 

that  is  the  equivalent  mass  from  the  moment  or 
aspect  is  1/2  the  mass  of  the  total  charge. 


EQUIVALENT  MASS  It  is  convenient  in  deriv- 

OF  ing  the  energy  equation, 

THE  RECOILING  PARTS     to  express  the  Kinetic 

Energy  of  the  recoiling  parts 
in  terms  of  the  velocity 

of  the  projectile. 

Neglecting  m  ds  small  as  compared  with  mr 
2 

and,  neglecting  the  recoil  brake  reaction  as  small, 
we  have,  by  the  principle  of  linear  momentum, 

f    ™  \   (        \ 
mr  V  =  (m  +  -  )  v  (  approx.) 


40 


hence        (m   *   ) 


Therecoil  Energy,  becomes, 


?  »r  v*  =  i  I va        (23) 


and  therefore  the  equivalent  mass  of  the  recoil 
ing  parts,  in  terms  of  the  velocity  of  the 
projectile,  becomes, 

(m  +  JL  )« 
2 


ENERGY  EQUATION         The  mechanical  work 

expended  by  the  gases  of  the 
powder  charge  in  the  bore  is 
equal  to  the  external  work  ex- 
erted on  the  projectile  and 

gun,  plus  the  Kinetic  Energy  given  to  the  gases 
themselves,  plus  the  heat  energy  lost  in 
radiation  through  the  walls  of  the  gun. 
If 

W  =  the  Potential  Energy  of  the  Gases  at  any 

instant. 
P|,j=  the  total  reaction  exerted  on  the  breech 

of  the  gun. 
P  =  the  total  reaction  exerted  on  the  base  of 

the  projectile. 
X  =  the  displacement  of  the  gun  measured  in  the 

direction  of  its  movement. 
x  =  the  displacement  of  the  projectile  measured 

in  the  direction  of  its  motion. 
E  =  the  Kinetic  Energy  of  the  powder  charge. 
Q  =  the  loss  of  heat  due  to  radiation. 


41 


J  »  the  mechanical  equivalent  of  heat  =  778  ***  lb* 

B.  T  .  U  , 

Then  for  the  energy  equation  of  the  powder  gases,  ire 
have, 

-Pb  dX  -P  dx  *  d(E  +  W)  *  JdO      (1) 
hence 

-dW  »  Pb  dX  +  P  dx  +  dE  +  JdQ.      (2) 
that  is  the  loss  of  the  potential  energy  of  the  gases, 
due  to  a  differential  expansion  goes  into  mechanical 
work  (PjjdX  +  P  dx  +  dE)  and  radiation  JdO. 
Further  by  (19),  (23)  and  (21), 


P  dx  =  d[y(m"va)) 


pbax 


AS  =  tlf  (5  v»  )] 

0 


so  that 


Further,  in  terms,  of  a  hypothetical  mean 
pressure  Pm(over  the  cross  section  of  the  bore) 

equation  (3)  may  be  expressed  in  terms  of  the 
travel  up  the  bore  u,  (i.e.  the  relative  displace- 
ment between  the  gun  and  projectile). 


where 

(6) 


(6) 

jec 
the  bore  of  the  gun,  approximately  since 


dv 

where  v  --—  =  the  acceleration  of  the  projectile  up 
du 


42 


dv   dx     dv      dv 

l~ 

dx 


v  dV  =  df  dx  +  dx   =  d?  l~TdT>  and  dX  is 


pared  with  dx,  and 

m"  =  the  equivalent  mass  of  the  projectile 
which  takes  care  of  its  angular 
acceleration  as  well  as  the  rifling 
friction,  see  equation  (19). 


EXPANSION  It  will  be  assumed,  that  the 

OF  expansion  of  the  gases  due  to 

POWDER  GASES       the  combustion  of  the  powder 

charge  obeys  the  law  of  a  perfect 

gas.   Hence, we  have, 

PV  =  RwT 
where  p  =  the  Intensity  of  Pressure  exerted  by  the 

gas Ibs/sq.  ft. 

V  =  the  volume  of  the  gas.  (cu.  ft.) 
w  =  the  weight  of  gas  (Ib.) 
R  =  a  coefficient  (ft.  Ibs.  per  Ib .  gas.) 
T  =  absolute  temperature  reached. 
Further,  with  a  perfect  gas,  the  internal 
energy  of  the  molecules  of  the  gas  is  entirely  in 
a  Kinetic  or  Vibratory  form,  and  therefore,  is 
directly  proportional  to  the  temperature. 
Hence,  we  have, 

dQ 
dQ  =  cwd  T  and  c  =  — 

where  dQ  =  the  heat  required  to  raise  the  gas  for 

a  change  of  temperature  dT. 
c  =  Specific  heat  or  the  heat  required  to 
raise  one  Ib .  of  gas  one  degree  of 
temperature  at  the  temperature  considered. 
We  are  concerned  especially  with  the  expansion  of 
a  gas  at  constant  pressure  or  at  constant  volume  or  a 
combination  of  the  two. 


43 


pdV 
Hence,  dQ.  =  wCp  dT  =  Cp  =--  —   at  constant  pressure 


Cv 

dQ  »  wC  dT  = 


R         at  constant  volume. 

If  the  volume  and  pressure  vary  together,  then, 

we  have  the  sum  of  the  partial  variations,  above, 
that  is, 

dQ  »—-  (C  p  dV  +  Cv  V  dp) 

Rw   " 

but  since,  dT  =  —  (p  dV  +  V  dp) 
R 

we  have,  dQ  =  CywdT  +  °P  '  °v  p  dV 
and  Q  _£ 

a  =  wCv/dT  +  p  R  v  —  /p  dv 

This  relation  can  be  interpreted,  physically 
immediately,  since  the  internal  energy  being  entirely 
of  a  kinetic  or  vibratory  form,  must  be  proportional 
to  the  change  in  temperature  at  constant  volume  other- 
wise additional  heat  must  be  added  for  the  external 
work.  Hence  wCy/dT  measures  the  molecular  kinetic 
energy.   Considering  an  expansion  at  constant  pressure, 
the  total  heat  required  is, 

Q  =  V  4-pdV 

•I 

where  U  *  the  internal  energy  =  wCy  dT. 

Since  the  heat  is  added  at  constant  pressure, 
we  a  Iso  have, 

pdV  =  wR  dT 

But  the  heat  added  at  constant  pressure  is, 
Q  »  wCp  dT 
hence,  substituting  in  the  total  heat  equation, 

wCp  dT  =  wCv  dT  —  w  R  dT 

^Dd       R         _R  _ 
Cp  -  Cv  '  T-  or  J  - 


P  -  <-v 


44 


If  now  the  specific  heats  Cv  and  Cp  are  assumed 
constant  for  the  range  of  temperatures  during  the  ex- 
pansion of  the  powder  gases,  we  have, 

Cp  -  Cv 
Q  =  *CV(T  -  TJ  +  -^ W   (7) 

where 

If  is  the  external  work  performed. 

Tt  is  the  initial  temperature. 

Neglecting  the  loss  of  heat  by  radiation  as  small, 
we  have  practically  an  adiabatic  expansion  in  the  bore 
of  a  gun;  that  is, 

-  C, 


wRT 
Since  p  =  -=-,  dividing  by  T,  we  have 

Cv  "  +  <Cp  -  Cv)  |L  0 


and  if  we  let 

Cp 

—  =   "    **«"   j        X"  ./v 

dV 


Cp          <jx 
-i-  =  n,  then  —  +  (n-1)^-  =  0 
uv 
and  , 


It  '          V 
Therefore, 

T-  .  c^)"-1 

Ta    V  ; 

Now  for  an  adiabatic  expansion,   Q.  =  0  and   therefore, 
(eq.7)  becomes, 

RwCv(Tt   -  T)  T, 

»  =  -7; «   — (Tt  -  TJ 

Cp  -  Cy  n  - 


hence 


7  n-1 


=  wCv  J  Tt  [1  -  (•—-)    1   (8) 


This  equation  1*  in  convenient  form  since  it  it 
in  terms  of  the  initial  and  final  volume  in  the  bore. 

The  equivalent  length  of  the  powder  chamber  of 
the  gun  in  terms  of  the  area  of  the  bore,  becomes, 

V, 

Vi   uo 


0.786  d*  so  that 


V    u0+  u 


In  terns  of  the  displacement  up  the  bore  the 
work  of  the  adiabatic  expansion  of  the  gases,  becomes, 

W  »  wCv  J  T,.  f  1  -  (  -  —  )"  "  l]   (ft  -  Ibs) 


1   (ft  -  Ibs.)  ) 

)  (10) 

* 

i  (-    •   ) 
n  -        uo  *  u 

Here 

w»  weight  of  gases  (Ibs.) 

W  «  external  work  performed  during  the 

(adiabatic)  expansion  (ft.  Ibs.) 
Cv*  specific  heat  for  constant  volume  (B-  T.  U.  per 

Ib.  per  deg.  ) 
Cp=  specific  heat  for  constant  pressure  (B.  T.  U. 

per  Ib.  per  deg.) 


T,=  initial  temperature  "(degrees) 

Vt»  initial  volume  (i.e.  volume  of  powder 

chamber) 

V«. 

u0  =  _±  -  where  0.7854d*  •  area  of  bore 
0.7854d'« 

u  =  displacement  of  projectile  up  the  bore 
J  =  mechanical  equivalent  of  heat  (•  778  ft.  Ib. 
per  B.  T.  IT.) 


46 


R»  (C   CV)J=  the  gas  constant  (ft.  per  degree) 

From  this  equation  we  may  deduce  the  differential 
equations  of  velocity  and  powder  pressure  for  the 
movement  of  the  projectile  up  the  bore,  provided 
we  know  the  manner  of  burning  of  the  powder  gases, 
etc. 

The  energy  equation  therefore  becomes:- 

2  wRT*  (i  .  .!£..  )n  -  l  =  ."  ! 1  •  v2  11 

'  n  -  1      u0*  u  ;  mr    3     J  (11) 

From  which  we  may  determine  v  in  teras  of  u. 
The  factor  dg  which  represents  the  loss  due  to 

beat  radiation  must  be  determined  by  experiment. 

Based  on  the  energy  equation  (or  its 
derivation,  the  force  equation  of  the  motion  of 
the  projectile  in  terms  of  the  displacement  up  the 
bore)  various  interior  ballistic  formulae  have  been 
derived  differing  in  the  method  assumed  as  to  the 
combustion  and  expansion  of  the  charge.   The 
formulae  of  Ingalls  and  Hugoniot  have  been  used 
by  our  Ordnance  from  time  to  time  especially  in 
ballistic  calculations.   In  recoil  design,  however, 
rough  approximations  are  sufficient  since  the 
manner  of  combustion  has  small  effect  on  the 
recoil.   The  formula  of  Leduc  is  sufficiently 
condensed  with  sufficient  approximation  to  be 
admirably  suited  for  recoil  design. 

TOROUE  REACTION  It  is  important  in 

OF  the  design  of  traversing 

THE  PROJECTILE        gear  for  guns  shooting  at 

high  angles  of  elevation  to 
compute  the  average  torque 
reaction  of  the  projectile  upon  the  gun. 
Let 

w  s  ang.  vel.  of  projectile  at  any  point  in  the 
bore  (rad/sec) 


47 


v  =  linear  velocity  in  bore  along  X  axis  (ft/sec) 
r  =  radius  of  bore  or  of  projectile  (ft) 
Pjj  =  powder  pressure  at  base  of  projectile  (ibs) 

f  =  normal  reaction  of  rifling  groove  (Ibs) 

T  =  torque  on  projectile  (Ib.  ft.) 
6  =  angle  of  rifling  grooves  with  XX 
I  =  mk*  =  moment  of  Inertia  of  Projectile 
IS  =  angle  turned  by  projectile 
Then 

u  =  v  tan  e 


d*0  =  tan  e  dgx   (21) 
dt*     r    dt2 

hence 

,  d  0   mk*     „  d*x 

T  =  nk* =  —  tan  0  — -   ,  ^. 

dt*   r        dt*   (12) 

but 

Tt2"    m    (approx.) 

hence  T  =  ~  tan  0  Pb  (Ib  ft.)   (13) 

2*r 
where  tan  0  =  


p  being  the  pitch  of  the  rifling  at  the  point 
considered . 

From  equation  (23)  we  see  that  the  torque  is 
proportional  to  the  powder  reaction  on  the  projectile, 
and  the  "slope"  of  the  rifling  grooves,  the  steeper 
the  grooves  being  the  greater  the  torque  reaction 
with  a  given  powder  pressure. 

Further,  if  the  rifling  pitch  is  made  constant 
throughout  the  greater  part  of  the  bore,  the  torque 


48 


varies  as  the  powder  pressure  curve  along  the 
bore  and  therefore  is  a  maximum  at  the  beginning 
of  the  travel  of  the  projectile. 

For  the  average  torque,  we  have, 

d»0 
Tav  s  »k*  (  -  ~\v 

u  t 

The  moment  of  inertia  of  the  projectile 
may  be  roughly  evaluated  by  assuming  a  solid 
cylindrical  projectile:- 

If     C  *  mean  or  equivalent  length 

D  =  density  or  weight  per  cu.  ft. 


DC  r  DC 

-  '  2  n  r  dr  =  ~ 


DC 
but  m  *  -— 


hence  -E£-nr*k-*  -—  -S~   and  k«  «-^-  ; 
g         2 

k  -  0.707  r 

Further,  it  is  customary  to  designate  the 
"rifling"  as  a  "twist"  of  1  turn  in  "g"  calibers. 
Therefore,  if  w,e  let, 
Twist:  =  1  turn  in  "g"  calibers 

Time  of  travel  in  bore  (approx.)  3  t  »  3/2  - 

v 

Radius  of  gyration  of  projectile  »  k  *  0.7  r 

Number  of  rev.  per  sec.  =  n 

Then, 

n  g  2  r  =  v  ) 

an£  )  at  the  nuzzle 

d0        HV  ) 

——  *  2nn  =  - 

dt        gf   d0        } 
Therefore,  since  T...t  =  mk(r  —  )  , 

dt   m 

T  -r—  -  mx  0.49  r*  - 
2v  gr 

and 

rev  " 

T  »  i.05---r  (24) 

ug 


49 


which  gives  the  average  torque  reaction  on  the  gun  due 
to  the  angular  acceleration  of  the  projectile. 

Since  the  slope  of  the  rifling  grooves  is  small, 
we  may  roughly  assume,  that,  a 

fr  =  T   and  f  =  1.05  -      (35) 

ug 

which  gives  the  mean  pressure  on  the  rifling  band. 

It  is  of  interest  to  compare  the  maximum  torque 
reaction  to  the  average  torque  reaction  in  an  actual 
gun. 

Type  of  Gun:  240  m/m  Howitzer 

Muzzle  Velocity:  1700  ft/sec. 

Weight  of  Projectile:  356  Ibs. 

Max.  Powder  Pressure:  32000  Ibs/sq.in. 

Rifling  =  1  turn  in  20  cals. 

Travel  up  bore  =  13.33  ft. 
Then,  for  the  max.  powder  reaction,  we  have, 

Pbmax.  =  32000  x  0.7854  *  -  =  2,242,000  Ibs. 
and  for  the  rifling  slope, 


hence,    for   the   max.    torque   reaction, 

9  45 
T     =  _:  —  x  0.157  x  2,242,000  =  69,500  Ibs.  ft. 

48 
where  as  for  average  torque,  we  have, 

u 

1.05  x  356  x  1700  x  9.45 

T  „  „  =  -  —  _  =  49600  Ib.  ft. 
av    13.33  x  32.2  x  20  x  24 

Therefore,  the  ratio  of  max.  torque  to  tne 
average  torque  becomes, 

Tmax  _  69500 
f  av  "  49600  = 

Due  to  the  short  time  action  of  the  travel  up  the 
bore,  the  effect  on  the  traversing  gear  depends  upon  the 
average  torque  rather  than  the  maximum. 


50 


MUZZLE   BRAKE 

GENERAL  The  muzzle  brake  consists  of  curved 

DESCRIPTION    vanes  secured  to  the  end  of  the  muzzle 
upon  which  a  portion  of  the  powder 

gases  are  deflected  in  the  second  part 
of  the  powder  period  after  the  projectile 
has  left  the  muzzle.  The  gases  are  deflected  somewhat 
to  the  rear,  and  we  have  a  forward  reaction  due  to  the 
change  of  momentum  of  the  gases,  which  materially  checks 
the  recoil.   The  design  and  best  arrangement  of  vanes 
requires  a  considerable  experimental  investigation  and 
it  is  merely  proposed  here  to  outline  certain  general 
limitations  based  on  an  elemantary  theory. 

ELEKANTARY        If  it  were  possible  to  calculate 
THEORY        the  mass  of  gas  discharged  through  the 
vanes,  as  well  as  the  mean  extrance  and 

exit  velocities,  the  reaction  on  the 
vanes  could  be  determined.  But  the 

•etbod  is  complicated,  since  the  pcmder  pressure 
after  the  shot  has  left  the  muzzle  falls  off  accord- 
ing to  a  complicated  function  of  the  time,  and,  con- 
sidering the  variable  volume  of  gas,  this  makes  it  dif- 
ficult to  approximate  the  mass  of  the  gas  as  a  function 
of  the  time.   Further  the  amount  discharged  through  the 
vanes  depends  upon  the  initial  mean  muzzle  velocity  of 
the  gases,  the  caliber  of  the  fcore  and  the  entrance 
areas  to  the  vanes,  as  well  as  the  variation  of  muzzle 
velocity  of  the  gases  against  time.  We  see  therefore 
to  approximate  roughly  the  problem  from  a  theoretical 
point  of  view  would  require  an  elaborate  analysis  com- 
bined with  a  long  and  eleborate  experimental 
research. 

Let 

w  »  weight  of  projectile  (Ibs) 

5  *  weight  of  total  charge  (Ibs) 
v  »  nuzzle  velocity   ft/sec. 


51 


Vo  =  velocity  of  recoiling  parts  when  the 

projectile  leaves  the  muzzle   ft/sec. 
to  =  time  for  shot  to  reach  muzzle. 

T  =  total  time  of  powder  period. 
Pjj  =  total  pressure  on  breech  due  to  powder  gases 

(Ibs) 
Wr  =  weight  of  reeoiling  parts 

vw  =  mean  velocity  of  gases  after  free  expansion 

(ft/sec. ) 

T 
/  Hdt  =  impulsive  reaction  on  vanes  (Ibs) 

„  charge  through  vanes 

Cw  =  ratio  of     s — - — r5 

total  charge 

Then,  without  vanes,  we  have, 


pb  dt  =       °  Pb  dt  +{pbdt         Total 
o  t0 

reaction  on   the 

Gun   during   the 
(w   +      -    )  v  Powder   period. 

Now  /   °   Pb   dt   =   

o  g 

and  since  the  powder  charge  has  a  mean  velocity  = 

v 

— —  when  the  projectile  leaves  the  bore, 

H 


±  W 

hence  /  pb  dt  =    v  +  -  vw 
o       £      g 

therefore,  WrVm  =  wv  *  w  vw 

as  we  should  expect  from  the  principle  of  the  con- 
servation of  momentum. 

With  the  muzzle  brake  acting,  the  total  reaction 


52 


during  the  second  period  (T  -t0),  on  the  gun  becomes, 

/T  Pb  dt  -  ;T8  dt 
' 


and  therefore  the  momentum  imparted  to  the  gun  be- 

wr (v    V  \ 
dt  »  -  <vra  -  o  ' 


comes,.?  _      T      wr  (v    V 
P  dt  - 


o 

If  it  were  possible  to  deflect  the  total  charge 

entirely  backward  and  maintaining  the  same  expansion, 
then,  for  the  total  reaction  on  the  gun  during  the 
expansion  period  of  the  powder  gases,  we  have, 


V 

since  the  change  in  velocity  *  vw  +  -   ft/sec 

ft 

Therefore,  the  momentum  given  to  the  gun  during  the 
powder  period  becomes, 

wv  -  w  v_. 


which  gives  the  impulse  imparted  if  we  had  muzzle 
brake,  with  the  sane  expansion  backward  as  forward 
through  the  vanes. 

With  the  same  expansion  to  the  rear  the 
maximum  possible  recoil  energy  that  can  be  absorbed 
with  a  muzzle  brake,  becomes, 


Aab 


2  w  w  v  vw 


*  »r 

and  the  maximum  possible  percentage  of  the  total  recoil 
energy  absorbed  by  the  ideal  muzzle  brake  becomes, 


53 


Aab    4  w  w  v  vm 


A     (wv  +  w  vw)* 

Since  w  vw  is  always  less  than  wv,  we  see  that 
even  with  an  ideal  muzzle  brake  and  complete  expansion, 
it  is  impossible  to  completely  check  the  recoil  energy, 
unless  greater  expansion  is  obtained  to  the  rear  than 
forward. 


The  total  gas  reaction  on  the  gun  due  to  the  com- 
bined expansion  and  deflection  of  the  gases,  is 
represented  by  the  impulsive  reaction, 

T  T 

/   R  dt  -/  Pjj  dt    in  a  forward  direction  (i.e. 
to       to         towards  the  muzzle) 

If  we  have  complete  expansion  of  the  gases, 
before  entrance  into  the  muzzle  vanes,  then, 

;T  pb  dt  -  ^  (v  .  y.  ) 

«*  '0        g  ^  W   U 

and  if  now  the  total  gases  are  deflected  entirely  back, 

then 

T       2  w  v.. 

r  R  dt  =  -  • 

* 


and  as  before, 


If,  however,  the  gases  are  accelerated  to  a 
mean  velocity  v1  before  entrance  into  the  vanes, 
then 

1*  V 

/T  Pb  dt  -  -  (v-  _  '  ) 
g 

and  further  expansion  takes  place  through  the  vanes 
to  the  maximum  value  vw  to  the  rear,  we  have 

/   R  dt  =  -  (v«  +  v  ) 


since  v1  +  vw  is  the  change  in  velocity. 

As  before  the  total  impulsive  reaction  on  the  gun, 
becomes, 

£H  at  -£  pb  at  .j  (,„  ,  J> 

Without  the  vanes,  the  reaction  on  the  gun  breech  be- 
comes,-       R       v 

{I  ?*  «  •  ;  K  -  I  ) 

and  with  the  vanes  the  reaction  on  the  breech  is  pro- 
bably different  and  modified  to, 

/T  Pfc  dt  =  *  (  V'  -  J) 

to 

since  some  expansion  probably  takes  place  within  the 
vanes,  themselves. 

Now  as  to  the  actual  reaction  obtained,  the 
ideal  brake  differs  from  actual  conditions,  essentially 
in  the  following  points:- 

(1)  Only  a  part  of  the  total  charge  can 
be  deflected  through  the  vanes. 

(2)  The  entrance  velocity  can  only  be  a 
component  of  the  actual  muzzle  velocity 
of  the  gases. 

(3)  Only  a  partial  expansion  of  the  gases 
can  take  place  before  entrance  into  the 
vanes . 

(4)  The  exit  velocity  can  not,  for 
practical  considerations,  be  entirely  to 
the  rear,  30*  from  the  rear,  being  like- 
ly the  maximum  angle  that  the  gases  can 
be  deflected. 

(5)  Only  a  very  small  expansion  can  take 
place  through  the  vanes  themselves;  of 
the  gases  passing  through  the  vanes  the 
total  expansion  is  small. 

In  consideration  of  (1)  (unless  the  vanes  are  ex- 
tended a  considerable  way  out)  the  higher  the  muzzle 


55 


velocity  the  less  the  total  charge  passing  through  the 
vanes.   It  has  been  found  experimentally  that  it  is 
useless  to  add  more  than  a  given  column  of  vanes, 
further  addition  of  vanes  having  very  little  effect 
on  the  reaction.   Further  the  first  one  or  two  vanes 
nearest  the  muzzle,  are  subjected  to  an  intensity  of 
pressure  practically  equal  to  that  of  the  gases  at  the 
muzzle.   Further  development  of  the  muzzle  brake  should 
be  directed  in  obtaining  greater  expansion  to  the  rear 
by  a  suitable  combination  of  vanes,  curvatures  of  same, 
etc. 

LEDUC'S  FORMULA         The  empirical  formula  estab- 
lished by  Leduc  is  especially  service- 
able and  sufficiently  accurate  for  a 
predetermination  of  the  reaction  of 
the  powder,  during  the  powder  period 
and  its  effect  on  the  recoil. 

Leduc's  formula,  assumes  that  the  velocity  curve 
of  the  projectile  during  its  travel  up  the  bore  follows 
that  of  an  equilateral  hyperbola,  with  parameters  a  and 
b,  that  is, 

If  v  =  the  velocity  of  the  projectile  at  any  point 

in  the  bore  (ft/sec) 

u  =  the  corresponding  travel  up  the  bore  (ft) 
a  and  b  being  parameters  of  the  hyperbola, 
then 

v  =  r-r: —  (ft/sec) 

b  +  u 

where  a  and  b  must  be  determined  by  the  elemantary 
principles  of  Interior  Ballistics. 

Determination  of  the  parameters  a  and  b:- 

When  u  is  made  infinite,   that  is  u  =  a 

and  v  -   a   -  a 


then  a 


56 


a  is  therefore  determined  by  considering  the  expansion 
in  a  gun  of  an  infinite  length.  c 

If  n  »  the  ratio  of  the  heat  capacities  (—*-  *1.4l) 
and  for  an  adiabatic  expansion  pV  =  k,         v 

Then  the  work  of  an  expansion  from,  initial  Volume 
Vt  to  final  Volume  V  ,  becomes, 


i 
=  /   p  d  V,  but  p  =  —  *  where  k  and  n  are  con- 


Now  when  Vf  becomes  infinite 

W  *  — —     *   t     (ft. Its.) 
n  -  1  Vt 

Since.  1  Ib.  of  water  *  27.68  cu.  in.  for  unit  density, 
k       1 


i  -  1  27.68"        Expansion  at  Unit 


Work  for  an  Infinite 

Expansio 

Density. 

Weight  of  given  volume  of  powder  gas 
Weight  of  same  volume  of  water 
and  if 

7C  »  the  given  volume  of  the  chamber   (cu.  in.) 

Vt  »  the  volume  of  1  Ib .  of  gas   (cu.  in.) 
then  ^ 

**    27.68 

A  *  »  — - —   per  Ib.  of  powder  gas. 

*a 

27.68 

hence  the  specific  volume  of  the  gas,  becomes, 

27.68 
V*a  '  "I" 


57 


Therefore,  the  work  of  expansion  of  1  Ibs.  of  the 
gas  to  oc  becomes, 

\f       An~l 

W 


1  27.68° 

Since  the  gas  evolved  is  proportional  to  the  weight 
of  the  charge  w,  and  a  =  v  for  an  infinite  expansion 
in  the  bore,  we  have 

waf 

w  E  A  *  ^  £or  a  compiete  expansion  of  w 


2g 

Ibs.  of  powder  gas, 

hence  a  »  /2gE  (-*—)*  A — 2 

Now  S  has  a  value  =  653  ft.  tons  roughly,  and  by 
experiment  "_lJ.  s  1/12  (approx.)  Taking  into  account 

the  various  losses,  it  has  been  further  found  ex- 
perimentally that  /2gE  =  6823  for  ordinary  good 
powder. 

Therefore,  the  parameter  "a"  becomes, 

a  *  6823  (— )t  ATI       ttf 

W 

27.68 
Now  A  a  -n 


but  with  a  powder  chamber  Vc,  loaded  with  w  Ibs. 
of  powder,  the  specific  volume  of  1  Ibs.  of  powder 
evidently  becomes, 

t^ 
''ta  *  w   assuming  complete  combustion  of  the 

charge, 

(2) 

that  is  the  density  of  loading  may  be  defined  as  the 
ratio  of  the  weight  of  the  charge  to  the  weight  of  a 


58 


volume  of  water  sufficient  to  fill  the  powder  chamber, 

Hence  the  parameter  a  becomes, 

i         __i 
a  =  6823  (-£-y  ,27.68  w.ta 

H     f        \          y  / 

vc 

To  evaluate  the  parameter  b,  we  must  consider  the  ac- 
celeration of  the  projectile,  and  the  reaction  of  the 
ponder  gases  on  its  base,  during  its  travel  up  the 
bore. 

The  acceleration  up  the  bore,  becomes, 

jiv      (b  *  u  )a-  av  K     a2  bu 
V"= 


(b  +  u  )'  (b  *  u)3   (3) 

since 

3.  V 

v  =  fr4.u    from  Leduc's  formula,  hence  the 
pressure  against  the  projectile,  for  a  displacement 
v,  becomes, 

w   aabu 
P  =  — 


g  (b+u)2  (Ibs) 

Further  the  maximum  pressure  occurs,  when  T— 
i.e.  when, 


]         .4       .3 
=  -3u(b  +  u)   +  (b+u) 


(b+u)4  and  u  =  -   (ft) 


that  is  the  maximum  pressure  in  the  bore  occurs  at  a 
displacement  equal  to  one  half  the  parameter  b  or  the 
parameter  b  -  twice  the  displacement  of  the  maximum 
powder  reaction  in  the  bore.   We  have,  therefore, 

substituting  v  =  °   in  (3) 
2 


59 


P    '        -    (Ibs)   (4) 


The  mean  powder  reaction  on  the  base  of  the 
projectile  during  its  travel  up  the  bore,  becomes, 


where 

VQ  =  the  muzzle  velocity   ft/sec. 

UQ  =  the  total  travel  up  the  bore   (ft) 
The  pressure  against  the  projectile  when  the  shot 
is  about  to  leave  the  muzzle,  becomes, 

a  bu 


Hence  to  determine  the  parameter  "b"  we  have  the 
following  equations:- 

4   w  a      . 
m   27  g  D 


f°-* 


\  where  Pm  VQ  and  UQ  are 
*   a*^_     (  known. 


2 

wv 


and   PQ  Pe,    a   and  b    are 
au 
v   :       b+tl  (  unknown. 

) 

Hence   a   solution   is  possible:-      If   A^  =   Area  of  bore 
and   Pm  =   a   given  property   of   the   powder   used 

Pm   =    30,000    to   33,000   Ibs/sq.in.    usually. 

Pm  =  p^  A   for    the  max.    powder  reaction. 
Substituting  atl 


e        2g      (b+u0)2 


o 

2 

(Ibs)   hence   a     = 


60 


P  .  4  »  £   (its)  bence  a" 
•   27  g  b 


4  w 
Equating,  we  have, 

2  Pe(b  *  uo)»   _  27  b  Pm 


hence 

27 
b»  +  2  b  u0  *  u*0  = 

o 


27 

t 

e 


+  12  -  3-  —  )  v  b  +  u20  =  0 

8   P. 


Solving,  we  have, 


(9    2?   m  1  ,       S<9 

"(2  "  8"  Pi  >  "o   !  /(2  '  g-  F* •  '   '  o  - 


(ft.)  (7) 

which  determines  the  parameter  b,  in  terms  of  the 
travel  up  the  bore,  the  given  maximum  powder  reaction 
and  the  mean  powder  reaction,  being  determined  from 
the  muzzle  velocity  and  travel  up  the  bore. 

To  completely  determine  the  velocity,  powder 
force,  and  time  against  the  travel  up  the  "bore,  we 
have 


— aa-    (ft/sec) 

b  +  u 


61 


P  = 


g   (b+u)3 
and  the  corresponding  time  of  travel,  becomes, 

,  du    (b+u) 
*  -  /  y-  '   (au)   du 

b  ,        1 
=  a  l°#e  u  *  a  u  *  Constant 

Now  when  u  =  0,  t  =  0  and  loge  u  =  -  a,  and  the 
constant  cannot  be  evaluated  without  making  some  as- 
sumption.  Since  the  initial  powder  reaction  required 
to  force  the  projectile  into  the  rifling  grooves  is 
large  and  the  displacement  u  »  «  ,  to  Max.  powder 
pressure  is  small,  we  can  reasonably  assume  the  powder 
reaction  constant  and  equal  to  the  maximum-  powder  re- 

action during  the  initial  travel  u  =  ^  .   Hence  as- 

2 
suming  the  maximum  powder  reaction  to  be  reached  at  the 

beginning  of  the  travel  of  the  shot  up  the  bore,  and 

then 

and   substituting   Praax>    from 


to  remain  constant  up  to  u  = 


hence  t.  =  v-~  (-)  (8) 


Sunstituting  in  the  previous  time  equation,  we 
have, 

'27"  b    b     b    b 

(-)  =  —  log  -  •»•  — -  *  constant 
a    a     2    2a 


62 


and 

Constant  =  —  ((/27  -  1 )  -2  loge  ^  ] 

a\  (2.098  loge  -  ) 
2  "a 

therefore 

2u    u 


=  -  (2.3  log  —  +  -  +2)  (approx.)  (9) 
a         b    b 

The  powder  reaction  on  the  breech  during  the 
travel  up  the  bore  is  somewhat  greater  than  at  the 
base  of  the  projectrle  due  to  the  inertia  resistance 
of  the  powder  gases  and  charge.   It  has  been  shown 
previously  that  the  breech  pressure  is  augmented 
over  that  at  the  base  of  the  projectile  by  either  of 
the  two  following  formulae:- 
v,  +  | 

Pfe  =  p  (Ibs) 

w 

or 

Pb  =  1.12  P   (Ibs) 

The  former  is  based  on  a  theoretical  assumption, 
and  gives  an  idea  as  to  the  change  in  the  pressure 

drop  from  the  breech  to  the  projectile  with  different 
ratios  of  powder  charge  to  weight  of  projectile.   The 
latter  is  entirely  empirical  and  it  appears  that  the 
ratio  of  the  weight  of  the  charge  to  that  of  the  pro- 
jectile has  no  effect  on  changing  the  ratio  of  the 
breech  pressure  to  that  at  the  base  of  the  projectile. 
Unfortunately  the  latter  empirical  value  is  somewhat 
limited  especially  for  extreme  ratio  of  the  projectile 
weights  but  is,  however,  reasonably  accurate  for 
ordinary  calculations.   The  former  is  more  or  less  in 
error  due  to  the  assumptions  made,  but  it  gives  the 


63 


characteristics  for  extreme  ratios.   Therefore,  for 
extreme  ratios  of  charge  to  projectile  weights,  the 
former  formula  should  be  used,  while  with  ordinary 
ratios,  the  latter  should  be  used. 


R  e  c  ap  i  t  ul  ait  ion  of  the  Various  Formulae 
Originating  from  LIDUC'S  Formula  - 

Let 

v  =  Velocity  of  projectile  up  "bore  (ft/sec) 

u  =  Travel  up  bore  (ft) 

v0  =  Muzzle  velocity  (ft/sec) 

UQ  =  Total  travel  up  bore  (ft) 

t  =  Time  of  travel  up  bore  (sec) 

to  =  Tine  of  total  travel  up  bore  (sec) 

W  =  Weight  of  powder  charge  (Ibs) 

w  =  Weight  of  projectile  (Ibs) 

Vc  =  Volume  of  powder  chamber       (cu.in.) 
A  =  Density  of  loading 

P  =  Powder  reaction  on  base  of 

projectile  (Ibs) 

P^  =  Powder  reaction  on  base  of  breech  (Ibs) 

PJH  =  Max.  Powder  reaction  on  projectile  (Ibs) 
Pe  =  Mean  Powder  reaction  on  projectile  (Ibs) 
A<j  =  Area  of  the  bore  (sq.in.) 

pm  =  Max.  given  powder  pressure     (Ibs/sq.in.) 
(from  30,000  to  33,000  Ibs/sq.in.) 

Given  :-   Pm  vo,  Vc,  w  w 
To  evaluate:  —  v,  P  and  t 


64 


27.68 
then,    A  = 


vc  (i) 

i   =    6823    (-)7   Al/l*  (2) 


(Ibs)  (3) 

(Ibs)  (4) 


U°     16  pe  16     pe 

v   =  -^—  (ft/sec)  <6) 

b   +  u 


P   =   - 


g        (b  +  u)»  (Ibs)  (7) 

w  a  2bu 

(b+u)3  (Ibs)  (8) 


w          a2bun 

P0v    =   1.12  -     2 

)3        (Ibs)  (9) 


t   =  -    (2.3   log   ^  t   -   *    2)      (sec)  (10) 

a  bo 


-    (2.3  log   ^      +  , 
a  bo 


t0   =   -    (2.3  log  +  ,—     +2)    (sec) 


3     ^ 
2     vo 


v  (approx.)    Uec)  (11) 


65 


DYNAMICS  OF  RECOIL         The  velocity  and  displace. 

DURING  THE  TRAVEL  OF     ment  of  the  recoiling  mass 

THE  SHOT  UP  BORE         with  respect  to  the  powder 

charge  and  projectile  is  ob- 
tained by  the  principle  of 

linear  momentum. 

Assuming,  one  half  the  charge  to  move  forward  with 

the  projectile  and  the  other  half  to  move  backward 

with  the  recoiling  parts,  we  "have, 

(  «r  *  \  >Vf  =  (  »  *  \  )v 

and   (wr  =  -  )Xf  =  (  w  +  -•  )x 
2  £ 

Now  the  absolute  displacement  of  the  shot  in  the  bore 
is  related  to  the  travel  up  the  bore  u,  by  the 
equation 

x  =  u  -  X 
Hence,  we  have, 


and 

w 
(w  +  -  )u 


Xf  = 


Since  w  and  w  are  small  as  compared  with  wr,  we  have 

for  a  sufficient  approximation 

w 
w  +  - 

Vf  =  2-  v      (ft. sec) 

w_ 


X   -iLlJl 

Af   -   '        *'    U         ft. 


The  equation  of  velocity  displacement  and  time 
of  free  recoil  during  the  travel  up  the  bore,  becomes, 


66 


Vf  = 


(wr  )      b  +  u 


(ft/sec) 


Xf-C- 


(ft) 


t  =  -  (2.3  log  —  +  e  * 
a         b    b 


With  constrained  recoil,  assuming  a  recoil  reaction 
X  we  have, 


dt 


Ubs) 


hence 


1  Pbd<-      Kt  =  v         *  pb  <' 

f  -r: —  v   but  /   -« vf 


therefore, 


Kt 


Vf  -  --  =  V  (ft/sec) 

mr 


Kt: 


2m, 


X  (ft) 


2u 


»  (2.3  log  ^  +  _ 

a         b    b 


+  2  (sec) 


In  the  several  equations,  it  will  be  noted,  that 
the  common  parameter  is  the  time  of  travel  up  the  bore 
in  the  gun.   Hence  if  for  various  values  of  u,  we  ob- 
tain correspondingly  values  of  time,  the  free  velocity 
and  displacement  is  obtained  for  the  given  time  and 
the  corresponding  effect  of  the  recoil  brake  during 
this  time  is  deducted  from  the  velocity  and  displace- 
ment respectively.  Further  it  has  been  tacidly  as- 
sumed that  the  powder  reaction  with  constrained  recoil 
is  the  same  as  with  free  recoil  at  the  same  time  in- 
terval.  This,  however,  is  not  strictly  true  since  the 


67 


powder  reaction  is  somewhat  modified  due  to  the 
slightly  different  motion  of  the  gun  with  constrained 
and  free  recoil  respectively.   The  effect,  however, 
is  entirely  negligible  as  compared  with  the  magnitude 
of  the  reaction  and  other  factors  involved,  even  with 
the  most  refined  measurements  and  analysis. 


EXPANSION  OF  THE  GASES  AFTER         The  manner  of 
THE  SHOT  HAS  LEFT  THE  BORE       the  expansion  of  the 
AND  ITS  EFFECT  ON  THE  RECOIL     powder  gases  after 

the  projectile  has 
left  the  bore  is 
very  difficult  to  calculate,  and  various  assumptions 

based  on  empirical  data  have  beeri  formulated,  for 
calculations  during  this  period. 

The  following  theory  though  imperfect  gives  an 
idea  as  to  the  manner  of  the  expansion  of  the  powder 
gases  in  the  "After  effect  Period". 

(1)     The  momentum  imparted  to  the  gun 

during  this  period  evidently  equals  the 
momentum  iwparted  to  the  powder  gases: 

n>r<Vf  *  Vfo>  '  S<vw  -  I  ) 

where   Vf  =  maximum  free  velocity  of  the  recoiling 

parts.    (ft/sec) 
Vf0  =  free  velocity  of  recoil  when  the  shot 

leaves  the  bore    (ft/sec) 
vw  =  mean  velocity  of  the  powder  gases  attained 

(ft/sec) 

i  =  mass  of  powder  charge    (Ibs) 
v  =  muzzle  velocity  of  projectile  (ft/sec) 
Since 

mr  Vj0  =(n  +  -  )v  we  have  mrVf  =  mv  +  m  vw 

In  other  words,  the  maximum  free  momentum  obtained  by 


68 


the  gun,  equals  the  sum  of  the  total  momentum  of  the 
projectile  and  the  total  momentum  of  the  powder 
charge. 

It  is  important  to  note  that  the  momentum 
relations  are  very  nearly  true  provided  we  are  able 
to  calculate  vw  the  mean  velocity  of  the  powder  gases 
and  can  neglect  the  small  effect  of  the  air  pressures 
exerted  on  the  gases. 

(2)     We  have  the  following  energy 

relations  due  to  the  expansion  of  the 

gases : 

(a)  Initially  the  gases  have  a 

Kinetic  Energy  =  *  (-)v* 
2   3 

(b)  The  work  of  expansion  of  the 
gases  in  expanding  from  the  pres- 
sure in  the  bore  when  the  shot 
leaves  the  gun  (i.e.  the  muzzle 
pressure)  to  the  atmospheric 
pressure,  becomes 

va 

We  =  /   pdV 
a. fee   y, 


ro 


where  Vo  =  volume  of  powder 
chamber  +  volume  of  the  bore  of 
the  gun. 

Va=  volume  of  gases  at  atmospheric 
pressure . 

(c)     The  final  Kinetic  Energy  of  the 
gases  may  be  approximately  assumed 
equal  to:  1  Sv^. 

2 

It  is  to  be  noted  that  the  final  Kinetic  Energy 
of  the  gases  is  difficult  to  calculate  due  to  the  di- 
vergence or  cone  effect  produced  when  the  gases  expand 
into  the  atmosphere   The  total  Kinetic  Energy  equals 


69 


the  sum  of  the  Kinetic  Energy  of  the  center  of  gravity 
of  the  gases  plus  the  relative  Kinetic  Energy  of  the 
gases  relative  to  the  center  of  gravity. 

From  a  series  of  experimental  tests  conducted  by 
the  Navy  on  the  velocity  of  free  recoil  with  guns  of 
various  caliber  it  has  been  ascertained  that  the 
momentum  effect  of  the  powder  gases  is  equivalent  to 
the  weight  of  the  charge  times,  a  constant  velocity  of 
4700  ft/sec. 

Assuming  the  divergence  of  the  spreading  of  the 
gases  to  be  similar  at  all  muzzle  velocities  ,  it  is 
possible  to  estimate  the  divergence  factor  and  then  in 
guns  of  very  high  muzzle  velocities  we  may  calculate 
the  maximum  free  velocity  by  multiplying  the  work  of 
expansion  by  the  divergence  constant  and  the  solving 
for  the  mean  velocity  of  the  gases. 

The  pressure  of  the  gases  rapidly  falls  to  the 
atmospheric  value  or  approximately  this  value,  before 
the  divergence  of  spread  of  the  gases  is  appreciable, 
hence  the  maximum  Kinetic  Energy  of  the  gases  will  be 
attained  at  approximately  atmospheric  pressure. 

The  change  in  Kinetic  Energy  of  the  powder  cases 
therefore,  becomes, 

•jSvw---v  =  change  in  Kinetic  Energy,  and 

the  work  done  on  the  gases,  equals  the  work  done  by 
the  external  pressures  po  and  pa  and  the  work  of  ex- 
pansion pdV.   Hence, 

vd 

p0V0  -  Pava  *  J    Pdv  ~  totai  work  done. 
vo 

To  allow  for  the  relative  Kinetic  Energy  due  to  the 
spreading  of  the  gases,  we  may  multiply  the  work  done 
on  the  gases  by  a  constant,  and  then  equate  this  value 
to  the  changes  of  the  translatory  Kinetic  Energy 
of  the  guns. 


70 


v  -n  v  +  f  a  PdV)  =-mv2  -im.,2 
o  vp  pa  va   J          2    w    *  q  v 
V 

vo 

where  K  =  the  divergence  constant  to  allow  for  the 
spreading  of  the  gases  at  the  muzzle.   Now  the  work 
of  expansion,  becomes, 

,Va         Po  vo  ~  Pa  Va 
We  =  /   T>  d  V  =  : 


'o 


where  the  expansion  exponent  k  =  1.3  approx.   Hence 
the  total  work  done  on  the  gases,  becomes, 

Po  Vo  -  Pa  Va     k  . 

Po  vo  -  Pa  v«  +  ; =  iT^I  (povo  ~Pava) 

K  *~"  X 

further,  since  prt  V ^  _  n  yk 

0  o   pa  va,  we  have, 


P     " 

rrr-fc"  v° " Pa  Va)  =  rri  p°  vo[1  -  (-^  )  ^  i 

PO 

Hence  the  energy  expression  reduces  to  the  convenient 
form, 


p  ^L_I_1_ 

K[r— — r  po  »o\l  -  '    /  k    >J=__va 
PO  2  3 

from  which  knowing  po»  VQ,  pa  m  and  v  enables  us  to 
immediately  calculate  vw,  the  mean  free  velocity  of 
the  powder  gases. 

To  evaluate  the  dispersion  constant,  to  take 
care  of  the  relative  Kinetic  Energy  of  the  gases 
after  expansion,  the  ballistic  data  of  the  155  m/m 
Filloux  gun  has  been  chosen,  since  assuming  a  mean 
velocity  of  the  gases  4700  ft/sec.,  calculated  and 
experimental  results  were  found  to  check  very  close- 
ly. 


71 


Weight  of  powder  charge  w  =  26  (Ibs) 
Volume  of  powder  chamber  S  =  1334  (cu.in.) 
Total  length  of  bore  u  =  186  (in.) 
Muzzle  velocity      v  =  2410  (ft/sec) 
Area  of  bore  Aj,  =  29.2   (sq.in.) 
Weight  of  projectile  =  96.1  (Ibs) 
Max.  powder  pressure  pm  =  35300  (Ibs/sq.in.) 
Mean  Powder  pressure  = 


wyg      _  _  19200  (Ibs/sq.in.) 

"  "e 
644  uAv 


Twice  Abscissa  of  maximum  pressure 

.27  "m    *i\  j.  -i/^ i         \ 2 IT  =  *^7  ^fl 

Muzzle  pressure  when  shot  leaves  muzzle 

_27      2  _u 27_          2          185.68   x    35300 

P°   =  ~4~  S      (e  +  u)3   P7n    "  T~     57.38X    (57.38    +   185. 68)3 

10140  Ibs/sq.in. 


we  have  then, 

0.3 
K  32.16  [— «  10140  x  U4  VIl  - 


0.3  V10140 

=    i  2  i  26  2 

•    -   x    26   x x  —      x      

2  4700          2  3  2410 

I 

Solving,   we   have, 

156   x    1Q«  K  V0   =    (287  -   25)  10 6    =    262   x    1Q« 


72 


Hence  K  =     =  0.430 


-  3.915  cu. 


Hence  the  energy  of  translation  is  but  43*  of  the  total 
Kinetic  Energy  of  the  gases  after  complete  expansion. 
Therefore  with  guns  of  numeral  ballistic  relations, 
we  may  estimate  the  mean  translatory  velocity  of  the 
gases  after  complete  expansion,  by  the  formula: 


o 


where  v  =  muzzle  velocity    (ft. see) 

w  =  weight  of  powder  charge   (Ibs) 
b  =  1.3  approx. 
.    pa  =  atmospheric  pressure  =  2116   (Ibs/sq.ft.) 

po  =  muzzle  pressure  of  powder  gases   (Ibs/sq.ft.) 

VALLIERS  The  hypothesis  of  Vallier  assumes, 

HYPOTHESIS     that  during  the  "after  effect  Period" 

in  the  powder  period  of  the  recoil,  that 
the  powder  reaction  on  the  gun  falls 
off  proportional  to  the  time.   That  is, 
If  Pob  =  the  total  breech  reaction  of  the  powder  gases, 

when  the  projectile  leaves  the  muzzle  (Ibs) 
to  ~  time  of  travel  of  the  projectile  to  the  muz- 
zle  (sec) 

tt  =  total  powder  period  (sec) 
P^  =  powder  reaction  on  "breech   (Its) 
t  =  corresponding  time  (sec) 
then 

?b  =  pob  -  c<t  -to)   VALLIERS  HYPOTHESIS 

where    p   _  -       p 

rob   Pb      rob 


c  = 


How  the  momentum  imparted  to  the  recoiling  parts  by 
the  gases  during  the  after  effect  period,  becomes, 


73 


tt 

Pfc  dt  »  mr(Vf  •  _  Vfo) 


where  Vfi  -  ^ax>  free  veiocity  of  recoil  at  end  of 
*  powder  period. 

Vfo  =  Free  velocity  of  recoil  when  the  shot 

leaves  the  muzzle. 


/  l  [Pob  f _t    (t  -  t0  ))dt  =  «r(Vf ,  -VfQ) 

1  Q  t          O 

Integrating,  we  have, 

P  b(  t-*t.) 

-i .  «r(V£I   VQ) 


2mr(Vf,    -Vfo) 
hence    t,    _   t    ,,   __L_J  -  H_  (sec) 


and 

pob 


C   = 


2(Vf.    -Vfo)mr 


Therefore   the   powder   reaction  during    the   after   effect 
period,    becomes, 


o 

Pb   =   pob — ^ — ; (Ibs) 

2(Vf.    -  Vfo)mr 


RECAPITULATION  Of  PRIKCIPLE  FORMULAS  OP 


INTERIOR  BALLISTICS  PERTAINING  TO 


RECOIL  DESIGN. 


The  velocity  and  displacement  of  the  recoiling 
parts  during  the  travel  of  the  projectile  up  the  "bore 
have  the  following  relations  with  the  velocity  of  the 


74 


projectile  up  the  bore  and  the  relative  displacement 
of  the  projectile  in  the  bore. 

(Weight  in  Ibs. ) 
If   m  =  massof  projectilev 00  .,  

oc  .  la 


m  =  mass  of  powder  charge 

mr  =  mass  of  recoiling  parts" 

v  =  velocity  of  projectile  (ft/sec) 

u  =  displacement  of  projectile  in  the  bore  from 

its  breech  position 
V  =  velocity  of  recoiling  parts   (ft/sec) 

X  =  free  displacement  of  recoiling  parts   (ft) 

then          =:           ra 
(m  +  -  )v     m  +  ; 
V  =  2 =  £_  v   Approx.  (ft/sec) 

ra         m 
m,.  +  •=         mr 


•  ,  ffl 

(m  +  -  )u      m  +  - 

X  - =  u   Approx.  (ft) 


The  pressure  on  the  breech,  in  terms  of  the  pressure 

on  the  base  of  the  projectile,  becomes 

If 

Pb  =  breech  pressure  (total)   (Ibs) 

P  =  pressure  at  base  of  projectile  (total)  (Ibs) 

m 
m  *  - 

P.,  =  P  =  1.12  P  approx.   (Ibs) 

m 

The  mean  pressure  in  the  bore,  becomes, 

I 

m  +  - 
4 

Pm  =  pb  ^ 1 — )       (Ibs) 


75 


For  building  up  the  energy  equation,  we  are  concerned 
with  the  various  equivalent  masses  of  the  moving  ele- 
ments that  the  powder  reacts  on  in  terms  of  the  major 
mass  of  the  projectile. 

The  equivalent  mass  of  the  projectile,  becomes, 
if   k  =  radius  of  gyration  about  its  longitudinal 
axis  (ft) 

p  =  pitch  of  the  rifling  " 

9  =  pitch  angle  of  the  rifling 


.          4k2tan2  9  ,      lh_ 
m  =  (1  +  -  )m  =  (1  +  -  )  a   |i«t 
P2  d  ' 


If  we  include  the  effect  of  the  friction  of  the  rifling 
we  have, 


sin  9  +  u  cos  6   4  tan  9  k2  ,     Ibs  . 
m  "  =[1  +  (  -  5  -  :  -  ^)  -  —  -  ]  m  (  -  ) 
cos  9  -  u  sin  €       d2 


The  equivalent  mass  of  the  powder  charge, 
for  the  energy  equation  =  35 

3    <i*I) 


for  the  momentum  equation  =  - 

- 


The  equivalent  mass  of  recoiling  parts  become, 
(m  +  =2 


ffli 


t 

The  differential  equation  for  the  motion  of  the  pro- 
jectile up  the  bore  becomes  in  terms  of  the  mean  pres 
sure  in  the  bore  Pm  and  the  relative  displacement  u, 

(m  +  |  )2 

.     _  2      m  .   dv 
Pm  =  [m  "  *•  -   +,  -  ]  v  - 

mw  du 


76 


If  W  »  the  potential  energy  of  the  gases  at  any  instant, 
we  have  further, 

-  dW  •  pm  du  *  JdQr 

where  fir»*  neat  lost  "by  radiation.  The  energy  equation 
for  the  expansion  of  the  gases,  becomes, 


where  w  »  (  pm  du  +  J  )  dOr   (ft/seo)(Adiabaticexpansion) 

w  »  weight  of  gases       (l*bs) 

cva  specific  heat  for  constant  volume  (B.T.  U. 

per  Ibs.  per 
degree) 

Op*  specific  heat  for  constant  pressure  (8.  T.  U. 
per  Ib.  per  degree) 


Ti  »  Initial  temperature  (degrees) 

Vi  *  Initial  volume  (i.e.  volume  of  powder  cham- 
ber) 


o  where  0.7854df  *  area  of  bore. 

0  . 


u  *  displacement  of  projectile  up  the  bore. 

J  *  mechanical  equivalent  of  heat  =  (  778  ft.  Ibs. 

per  B.  T.  17.) 

R  »   (Cp  -  Cv)  J  »  the  gas  constant   (ft.  per  degree) 
The   torque     reaction  of  the  projectile  in  travel- 
ing up  the  bore  becomes, 


77 


mv* 
1.05  -  r    (Ibs.ft) 


where   r  «  radius  of  tho  bore  (ft) 

v  =  muzzle  velocity  (ft/sec) 

g  »  number  of  calibers  per  revolution 

LEDUC'S  FORMULA         Leduc's  formula  gives  results 
sufficiently  accurate  for  recoil 
design.   The  formulas  derived  from 
it  are  compact  and  sufficiently 
short  to  "be  readily  used  in  ordinary 
practical  design.  These  formulas  have  been  used  in 
the  development  of  the  various  recoil  formulas  in  the 
subsequent  chapters.  ?or  recoil  or  gun  design: 
let 

v  =  velocity  of  projectile  up  bore  (ft/sec) 
u  =  travel  up  bore  (ft) 

v0=  muzzle  velocity  (ft/sec) 

UQ«  total  travel  up  bore          (ft) 
t  =  time  of  travel  up  "bore        (sec) 
tQ3  time  of  total  travel  up  bore    " 
w  =  weight  of  powder  charge       (Its) 
w  =   "of  projectile 
Vc  'Volume  of  powder  charge        (cu.  in.) 

A  =  Density  of  loading 

P  »  powder  reaction  on  base  of  projectile   (Its) 
Pb  =  "  reaction  on  base  of  breech          " 
Po}jsPressure  on  the  projectile  when  the  shot 

leaves  the  muzzle        (Ibs) 
Pm  =Maximum  powder  reaction  on  projectile  (Its) 
Pe=  Mean  powder  reaction  on  projectile      " 
A(j=  area  of  bore  (sq.in.) 

Pm=  Maximum  given  powder  pressure  from 

25000  to  33000  Ibs/sq.in.         (Ibs/sq.in.) 
Given, 

pm,  VQ,  Vc,  w,  w  and  UQ 


78 


To  evaluate:-  v,  P  and  t,  then, 

27.68  w 
(1)     A- 

__   (2)     a  =  6823  (->!  A*  " 

w 

(3)     Pm  =  PmAd   (Ibs) 


2 
W  V 

(4)     Pe  =  — £—   »      g  =  32.16  ft/sec-2 


. 

(6)  v  =  ^L-    (ft/sec) 

b+u 

w  a2bu 

(7)  P  = (ibs) 

g  (b  +  u)3 

w  aabu 


(9) 


g  (b+uQ)3       (Ibs) 

vr'iC  ;O  «**a  f- 
- 


(10)    t  =  -  (2.3  log  —  +  £  +2)   (sec) 
a         b     b 

b         2U0   U0 

to  =  ~  (2.3  log  — -  +  —  +  2)   (sec) 
a         b     b 

(ID  t0 « -  H® 

2  VQ   approx. 

The  equations  of  velocity,  displacement  and  time 
of  free  recoil  during  the  travel  up  the  bore,  becomes, 


79 


w 
w  +  - 

p       a*j 

Vf  =  ( ^)(     )     (ft/sec) 

wf   b  +  u 

»  *5 

Xf  =  ( -)  u          fft) 

wr 

:*jp»-~ee  •  •         •  .-•• 

20   u 

,  =  1  (2.3  lo?  r-  +  5-  +  2)   (sec) 
a 

With  constrained  recoil,  assuming  a  recoil  reaction  K 

Kt 

V  =  V*  -  —  (ft/sec) 


X  =  Xf  -  —  (ft) 

2mr 


t  =   (2.3  log    +  £  *  2)    (sec) 
a         bo 

Theexpansion  of  the  gases  after  the  projectile 
leaves  the  hore  causes  an  additional  recoil  effect. 
The  hypothesis  of  Tallier  assumes  the  powder  reaction 
to  fall  off  proportionally  with  the  time.  On  this 
assumption: 

If 

Vf  ~  the  velocity  of  free  recoil  at  the  end  of 
the  powder  period. 

Vfo  =  the  velocity  of  free  recoil  when  the  shot 

leaves  the  muzzle. 

p 

tt  and  t0  the  corresponding  terms  --- 


(sec) 
ob 


.'   "  -  P'01> 


2(Vfl  _  V)Br    (IT,.) 


80 


»  >y 

THE  PARABOLIC  TRAJECTORY       The  nucleus  of  exterior 

ballistics  is  the  differential 
equations  of  the  parabolic 
path  of  a  shot  projected 
in  a  vacuum.  These  equations 

then  nay  be  modified  for  air  resistance  and  gyroscopic 

deflections  due  to  the  angular  momentum  of  the  pro- 

jectile and  air  reaction: 

Let  x  and  y  be  the  horizontal  and  vertical  coordinates 

of  the  trajectory. 

n  *  the  mass  of  the  projectile. 

Vo  =  the  muzzle  velocity. 

t  »  the  time  of  flight. 

0'  =  the  angle  of  elevation  from  the  horizontal 

of  the  axis  of  the  bore. 
0  =  angle  of  elevation  of  the  departure  of  the 

projectile  from  the  muzzle. 
e  *  the  increment  angle  or  "jump"  to  the  elastic 

deformation  of  the  carriage  and  the  move- 

ment of  the  gun  in  a  direction  not  along  the 

axis  of  the  bore.  01-  0 
r  =  angle  of  sight. 
Oaa  line  of  sight. 
L  *  range  to  given  target. 
LQ  =  horizontal  range  corresponding. 
•  =  striking  angle  from  horizontal. 
m1  =  angle  of  fall  from  line  of  sight. 

The  differential  equations  of  motion  give: 
d»x  d«y 


Integrating  successively,  we  have, 

~  »  V0  cos  0     ~  =  -  gt  +  V0  sin  0 
dt 

and 


.  \\ 


82 

^^^^^ 


x  =  V0  cos  0  t       y  =   -   -   +  VQ  sin  0  t 

<Q 

Hence        g     x2 

y  =  -  -  +  x  tan  0      (1) 

2  V2Q  cos2  0 

The  general  parabolic  equation  of  the  trajectory 
in  vacuum.   For  maximum  range,  x  being  a  function  of 
0,  we  have 

—  =  0,  when  y  =  0  in  (1) 
d0 

that  is,  2V2Q  V2 

x  =  sin"0  cos  0  =  — °-  sin  20 

g  g 

and 

dx  2V*Q 

— -  =  cos  20  =  0 

d0  g 


Hence,  cos  2  0  =  0  or  0  =  45° 

When  air  resistance  is  considered  maximum  range  for 
ordinary  guns  is  obtained  at  angles  which  may  be  from 
about  42  to  55  degrees.   If  x  and  y  are  the  coordinates 
of  some  target  point,  we  have  for  the  equation  of  the 
line  of  sight,  that 

y  =  x  tan  r 
Substituting  in  (1) 


X2 

«  COS20  + 


gx 

7""*    ..,     =  tan  0  -  tan  r 
2V_  cos*0 


and  ya  »  xa  tan  r  and  L  =  -  xa2  -  y 


which  gives  the  coordinates  and  ranges  in  terms  of 

the  muzzle  velocity,  angle  of  departure  and  angle  of  sight, 


CHAPTER   III 

EXTERNAL  REACTIONS  ON  A  CARRIAGE  DURING  RECOIL 
AND  COUNTER  RECOIL  - 
STABILITY  - 
JUMP. 

EXTERNAL  REACTION     The  external  reaction  during  re- 

coil may  "be  divided  into  two  primary 
periods;  that  during  which  the  force 
of  powder  pressure  on  the  recoiling 
parts  exceeds  the  restraining  force 

or  accelerating  period  and  the  retardation  period.  Again 
the  period  of  powder  pressure  may  be  divided  into  the 
period  of  the  shot  traveling  up  the  "bore  to  the  muzzle 
and  the  after  effect  period  of  the  powder  gases  ex- 
panding to  atmospheric  pressure. 

Considering  the  gun,  recoiling  masses  and  carriage 
as  one  sustem,  the  external  forces  are: 

(1)  The  pressure  of  the  powder  gases 
along  the  axis  of  the  bore  »  ? 

(2)  The  torque  reaction  due  to  rifling  *  T 

(3)  The  weight  of  the  recoiling  parts  »  Wr 

(4)  The  weight  of  the  stationary  parts*  W& 

(5)  The  balancing  reactions  exerted  by 
the  ground  or  platform  on  the  carriage 
mount  . 

If  we  sum  these  forces  up  into  X  and  Y  components 
and  let  Hr  equal  the  mass  of  the  recoiling  parts,  we 
have,  noting  the  mass  x  acceleration  of  the  stationary 
parts  of  the  system  is  nil,  ithat) 


«, 


dt* 

83 


34 


If  further,  we  assume  our  coordinates  along  and 
normal  to  the  axis  of  recoil,  we  have 


X  =  M, 


daxr 

dt2 


Y  =0 
Equation  (1)  may  be  written: 


(I1) 

(2') 


ZX  -  Mr 


d2x, 
dt; 


=  0 


Hence,  by  the  use  of  D'Alemberts  '  principle  regarding 
the  inertia  effect,  that  is,  mass  x  acceleration  re- 
versed, as  an  equilibrating  force,  we  reduce  the 
forces  to  a  system  of  forces  in  equilibrium. 

Thus  by  including  the  inertia  effect  of  the  re- 
coiling parts  as  an  additional  external  force,  the 
problem  is  reduced  to  one  of  statics. 

This  greatly  simplifies  the  procedure  of  ac- 
curately and  quickly  obtaining  certain  overall  effects 

in  stability  and  the  principal  reactions  throughout  a 
carriage. 
i  ;•:  irii«9  tea  if*t«fl  $riiiaa«?  ,Jejt.  sd:  -       ,ioO 

EXTERNAL  EFFECTS          Considering  now  the  external 
DURING  RECOIL       reactions  upon  the  total  system, 
(gun,  recoiling  parts,  and  car- 
riage proper)  including  the  inertia 
of  the  recoiling  -masses,  we  have 
the  given  forces  as  shown  in  figure  (1),  where 


Fig.  1 


85 


r 


P  =  Total  Powder  Pressure  along  axis  of  bore  at 

any  instant  of  Powder  Pressure  Period. 
Wr  =  Weight  of  recoiling  masses  "Mr"  . 
Wa  =  Weight  of  carriage  proper  (includes 
stationary  part  of  tipping  parts  ) 

H  £*_=Inertia  force  of  recoiling  masses 
r  d  t 


Ha  and  Va  =  Horizontal  and  Vertical  components 

or  equivalent  float  reaction 

7^  =  Front  Pintle  reaction  -  Horizontal  component 
assumed  zero  in  order  that  the  reaction  may 
be  determinate. 

B  =  Braking  force,  resultant  hydraulic  and  re- 
cuperator reaction  including  stuffing  box 
frictions. 

R  =  Guide  frictions  =  Ri  +  R2  in  diagram. 
Ka  =  Total  resistance  to  recoil  for  recoiling 

masses  equals  B  +  R  -  Wr  sin  J0  at  any  instant 
during  powder  pressure  period. 
Kr  =  Total  resistance  to  recoil  during  any  in- 

stant after  P  =  0. 

During  the  powder  press.ure  period,  we  have  for 
moments  about  A,  see  figure  (1) 


P(d  +  e  )  -  (Hr   1  )d  -  Wr  Lr  -  Wa  La  +  VbL  -  0 

d2x 
-  (Mr  7~7>id  *  Pe  ~  wrLr  ~  ffaLa  +  vbL  =  ° 


hence 


(3) 


In  like  manner  we  have  after  the  powder  pres- 
sure ceases 


(4) 


Now  considering  the  external  reactions  on  the 
recoiling  parts  alone*,  during  the  powder  pressure 
period,  we  have  figure  (2) 


86 


Fig.  3. 


hence 


P  - 


and 

when  P  »  0  figure  (3) 


d'x 


k  dt 
d'x 

— =  *  B  +  R  -  If-  sin  j 
dt 


(5) 


,t 


B  +  R  -  Wr  sin  0  =  Mr  ~ 


r 
dt 


(51) 


87 


Substituting  (5)  and  (51)  in  (3)  and  (4)  respectively, 
we  have 

K»d  +  Pe  -  WrLr  -  WaLa  +  VfcL  -  0   (6) 


Krd  -  VrLr-  WaLa  +  VbL  »  0        (7) 

Thus  the  external  effect  during  the  powder  pressure 
period  is  always  at  every  instant  equal  to  the  total 
resistance  to  recoil,  that  is,  the  sum  of  the  total 
braking  and  guide  friction,  minus  the  weight  component 
and  a  powder  pressure  couple  Fe  dependent  upon  the 
actual  total  powder  force. 

In  general,  e  is  very  small  and  usually  for  a 
first  approximation  the  powder  pressure  couple  can  "be 
neglected. 

Further,  for  constant  resistance  to  recoil 

Kr  -  Ka  -  K  -  B  +  R  -  VTr  sin  0      (8) 

which  is  the  average  external  effect  during  recoil  on 
the  total  system. 

As  shown  in  Chapter  VI  on  the  "Dynamics  of  Re- 
coil"    i 

K  «  -* — E £ — 

b  -  E  +  VfT  (9) 

where  Vf  *  the  maximum  free  velocity  of  the  recoiling 
parts,  that  is 

_  W4700  +  WQ 

wr  (10) 

W  *  Weight  of  powder  charge,  W  *  Weight  of  shot 
and  Wr  •  Weight  of  recoiling  parts 
v0  *  Muzzle  velocity  of  shot 
u  »  Travel  up  the  bore  in  inches 
b  *  Length  of  recoil  in  feet 

d  a  Diameter  of  bore  in  inches 

E  »  Unconstrained  displacement  of  recoiling  parts 
during  powder  pressure  period. 


88 


T  =  Total  time  of  powder  pressure  period. 

In  equation  (9)  note  that  E  =  Kt  Vf  T   and 

K 

o 


2  v 


Substituting  these  values  in  (9)  and  solving  for 
a  wide  range  of  artillery  material  and  thus  evaluating 
the  variables  as  a  function  of  the  diameter  of  bore, 
muzzle  velocity  and  travel  up  bore,  Mr.  C.  Bethel  has 
given  the  very  valuable  and  serviceable  formulae,  and 
accurate  to  one  percent. 

Mr  Vf  1 

=  —  =  -   -   - 


b  +  (.096+. 0003  d)  uvf 

vo          >*|»H 

This  formula  holds  only  for  constant  resistance 
to  recoil. 

It  is  important  to  note  that  the  "total  braking" 
sometimes  called  "the  total  pull"  is  not  in  general 
equal  to  the  resistance  to  recoil,  but  is  the  total 
resistance  to  recoil  plus  the  weight  component,  that 
is 

B+R=2Pa+ZPn+2Rs+2fig  =  K+Wr  sin  0    (12) 

where  2Pa  =  Total  recuperator  reaction 
ZPj,  =    "    hydraulic  reaction 
2Rs  =   "   stuffing  box  friction 

2R,<  =  Guide  friction 


K  =  T  M.  V, 


ir  vf 


(b  -  E  +  Vf  T) 


To  obtain  the  external  reactions  on  the  carriage 
mount,  it  is  convenient  to  know  d  in  the  previous 
moment  formulae  about  A,  in  terms  of  the  height  of 
the  trunnions  and  the  distance  between  the  trunnions 
and  a  line  through  the  center  of  gravity  of  the  re- 
coiling parts  arjd  parallel  to  the  axis  of  the  bore. 


89 


Let  H+  =  height  of  trunnions  above  the  ground 

distance  from  trunnion  axis  to  line  through 

center  of  gravity  of  recoiling  parts  and 

parallel  to  l>ore. 

moment  arm  of  K  about  A  nor. 

horizontal  distance  bet-ween  reactions  A  and  B. 

from  A  to  center  line  of 
trunnions. 
As  the  gun  elevates,  we  have  two  cases: 

(1)  When  the  line  of  action  K  passes 
above  A,  see  figure  (4) 

(2)  When  the  line  of  action  K  passes  be- 
low A,  see  fig.  (5) 


t 

s  = 


d  = 
1  = 
c  = 


-*-  K 


Fig.  4 


90 


r 


Fig.  5 


Fig.  5' 


91 


For  case  (1),  we  note  that 

h'  -  (d  sin  0  +  c  )  tan  0  =  d  cos  0 

but  s 

*'  =  Ht  +  ^I~0 
and  a 

E  +  _f d  sln  0  _  c  tan  0  =  d  cos  0 

cos  0   cos  0 

ht  cos  0  +  s  -  d  sin2  0-  c  sin  0  =  d  cos  0 
hence 

d  =  ht  cos  0  +  s  -  c  sin  0         (13) 
For  case  (2),  we  note  that 

h1  +  d  cos  0  =  (c  -  d  sin  0)  tan  0 
but 

n'  =  ht  *  — — ^ 
1   cos  0 


cos  0      cos  0 


2  2 

ht  cos  0  +  s  +  d  cos  0  =  c  sin  0  -  d  sin  0 
hence  d  =  c  sin  0  -  ht  cos  0  -  s       (14) 

If  W  =  weight  of  the  total  system  (gun,  recoiling 
parts  and  carriage  ),  we  have  for  moments  about  A 

*s  ^s  =  vr  Lr  +  wa  La      In  battery 

or  IB  terms  of  the  tipping  parts  =  Wt 

and  the  top  carriage  alone  (not  including  the  stationary 

parts  of  the  tipping  parts  =  Wa 

Ws  Ls  =  Wt  Lt  +  Va  La      In  battery 

where  Ls  =  distance  to  center  of  gravity  of'  total  sys- 
tem in  battery  from  A 

If  b  =  length  of  recoil,  and  0  the  angle  of 
elevation.,  and  Lg  -  distance  to  center  of  gravity  of 
system  out  of  battery,  we  have 

Ws  14  =  Wr  (Lr  -  "b  cos  0)+  Wa  La 

=  ¥r  Ly  +  Wa  La  -  ¥r  b  cos  0 
hence  TTS  L^  =  Ws  Ls  -  ^r  b  cos  0   Out  of  battery 


92 


Hence  the  external  reactions  at  A  and  B  on  the 
carriage  mount  become  in  terns  of  the  resistance  to 

recoil,  powder  pressure,  height  of  trunnions  and 
distance  between  trunnions  and  line  through  center  of 
gravity  of  recoiling  parts  parallel  to  axis  of  bore, 

For  low  angles  of  elevation, 
Taking  moments  about  A,  we  have, 
Vb  L  +  Kd  +  Pe  -  Ws  Ls  +  tfr(x  cos  0)=  0 
hence     Wg  Ls  -  Wr(x  cos  0)-  Kd  -Pe 


Pe  disappearing  for  a  finite  value  of  x  or  in 
other  words,  when  pe  is  used  Wr  x  cos  0  may  be 
neglected.   And  since  Va  r  Ws  +  K  sin  0  -  V^  or 
directly  from  moments  about  B,  noting  that  moment 
arm  of  K  becomes  d'=  d  +  L  sin  0  =  ht  cos  0  +  (L-c)sin  £l 

*  S 
we  have, 

Wa(L-Ls)  +  WP  x  cos  0  +  K(d+L  sin  0)  -  Pe 

V     3 

Va  L 

and  as  before  Pe  disappearing  unless  x  is  very  small. 
Obviously  Ha  =  K  cos  0  and  is  in  no  way  directly  ef- 
fected by  the  powder  force. 

For  high  angles  of  elevation,  the  moment  arm 
Kd  reverses,  and  d  and  d1  become  respectively, 

d  *  c  sin  0  -  ht  cos  0  -  S 
and 

d1  »  L  sin  0  _  d     See(fig.5) 
=  (L-c)  sin  0  +  bt  cos  0  +  S 
Now  taking  moments  about  A  and  B  respectively 
Wg  Ls  -  Wr  x  cos  IS  +  Kd  -  Pe 

b          L 
and 

W8(L-Lg)  +  Wr  x  cos  0  +  K(L  sin  0  -  d)  *  Pe 


93 


and  Ha  -  K  cos  0. 

For  design  use,  the  external  reaction  formulae 
be  conveniently  grouped. 
IN  BATTERY:  for  low  angles  of  elevation: 


j  r    Nvn*  cos  0  +  S  -  c  sin  0)  -  Pe      ) 

•»  g  Ul  a   ™         v 

)  V  "       "T~ 

)      WS(L-LS)  +  K(htcos  0  +(L-c)  sin  0  +S)+Pe   ( 

)   Ha  »  K  cos  0  ( 

for  high  angles  of  elevation: 
WgLg+  K(c  sin  0  -  ht  cos  0  -  S  )  -  Pe 

(  L  ) 

>      Ws(L-Lg)+K[(L-c)  sin  0  +ht  cos  0+S]  +Pe    ( 

(  Va»  )  (16) 

)   Ha  a  K  cos  0  ( 

OUT  OF  BATTERY:  for  low  angles  of  elevation: 
(      ya^s  ~  wr  k  cos  ^~  ^^ht  cos  0  +S-  c  sin  0  )  ) 

)      WS(L-LS)  +  Wr  b  cos  0+K(ht  cos  j0+(L-c)sin  0+S) 

)  v     ~        T""(17) 
c  ) 

)   Ha  =  K  cos  0  ( 


94 


for  high  angles  of  elevation: 
b  cos  0  +  K(c  sin  0-  ht  cos  0-  s 


WS(L-LS)+  Wrb  COS0  +  K(htcos  0*  (L-c  )sin#+S 


L 


(  Ha  =  K  cos  0  ) 

These  formulae  are  immediately  applicable  to 
platform  mounts  traversing  about  a  pintle  bearing  as 
well  as  field  carriages. 

In  platform  mounts,  the  horizontal  reaction  of 
the  platform  on  the  mount  is  usually  taken  at  the 
pintle  bearing  which  is  usually  located  in  the  front 
or  muzzle  end  of  the  mount.   Hence  in  place  of  Ha  we 
have  HO  =  K  cos  0.   The  reactions  V^  and  Va  remain 
the  same,  Va  now  being  the  reaction  of  the  platform 
on  the  traversing  rollers  of  the  mount.   Very  often 
V)j  is  divided  into  two  equal  vertical  oomponents  at 
the  two  ends  of  the  traversing  arc  of  the  mount,  and 
in  such  a  case  L  is  the  horizontal  distance  in  the 
projection  of  a  vertical  plane  containing  the  axis 
of  the  bore  from  the  pintle  reaction  to  the  traversing 
reaction,  that  is,  if  L  is  the  actual  distance  from 
the  pintle  to  the  other  end  of  the  traversing  arc, 
and  9  is  the  spread  of  the  arc,  then 

6 

L  =  L'  cos  - 
2 

In  a  field  carriage,  for  a  first  approximation 
we  may  assume  the  horizontal  and  vertical  reaction  to 
be  at  the  contact  of  spade  and  ground.   These  reactions 
are  obviously  Ha  and  Va  of  the  previous  formulae  and 
Vfc  is  the  vertical  reaction  of  the  ground  on  the 
wheels,  and  L  the  distance  from  the  wheel  contact  to 
the  spade  contact  with  the  ground.   For  split  trails, 


95 


Va  and  Ha  are  obviously  equally  divided  and  if  the  gun 
is  traversed,  a  horizontal  reaction  normal  to  the 
plane  of  Ha  an^  Va  is  introduced;  however,  this  re- 
action will  not  be  considered  until  later,  that  is, 
the  gun  will  be  assumed  at  zero  traverse. 

.A  closer  approximation  to  actual  conditions  in 
a  field  carriage  is  to  regard  H  as  acting  at  a  vertical 
distance  g  from  the  ground  line,  usually  when  from  1/2 
to  2/3  the  vertical  depth  of  the  spade  in  the  ground. 

The  equations  then  will  have  an  additional  moment: 

Hag  =  K  cos  0  g, 

which  is  substracted  from  the  moments  of  the  numerator 
in  the  expression  for  V^  and  added  to  the  mpments  in 
the  expression  for  Va-   The  general  equations  for 
field  carriages  are  then, 

for  low  angles  of  elevation: 

WSLS-  Wr  x  cos  0  -  K.(d  +  g  cos  0)-Pe 

vb  =  . 


W3(L-LS)  +  Hr  x  cos  0  +  K(L  sin  0  -  d  +  g  cos#)+pe 

Va  = 


Ha  =  K  cos  0 

d  -  ht  cos  0  +  3  -  c  sin  0 

and  for  high  angles  of  elevation: 


Y^L,  _  Wr  x  cos  0  +  K(d-g  cos  0)  -  Pe 

_.  *S  .  ^  _____  _______  ___ 

L 
WS(L-LS)  +  Wrx  cos  0  +  K(L  sin0-d+g  cos0)+  Pe 


Ha  =  K  cos  0 

d  =  c  sin0  -  ht  cos0  -  S 
where  Pe  disappears  if  Wr  x  cos  0  is  used  or  vice  versa. 


96 

Another  class  of  mounts  in  which  the  previous 

formulae  are  not  applicable,  are  known  as  pedestal 
or  pivot  mounts  used  on  Barbette  Coast  mountings  and 
for  naval  guns,  as  well.  These  mounts  are  attached 
to  the  foundation  by  bolts  on  a  circular  base  usualljr 
equally  spaced  around  the  circumference. 

With  such  mounts  the  question  of  stability  is  of 
no  consideration.   The  reaction  between  the  foundation 
and  mount  and  the  distribution  of  the  tension  in  the 
bolts,  may  be  obtained  approximately  by  considering  the 
base  of  the  mount  as  absolutely  rigid.   Then  on  firing, 
the  front  bolts  become  the  most  extended,  the  deflect- 
ions and  corresponding  stress  being  proportional  to  the 
distances  measured  from  the  back  end  of  the  base  along 
the  trace  of  the  intersection  of  vertical  plane,  con- 
taining the  axis  of  the  bore  with  a  horizontal  plane, 
to  the  perpendicular  chord  connecting  any  two  front  bolts 

Thus  if  L0,  L,  etc.  are  the  lengths  from  the  base 
end  to  the  perpendicular  chord  connecting  a  set  of  two 
bolts,  and  if  jo,  j,  etc.  are  the  deflections  of  the 
bolts,  we  have  j0:  jt:  jt:  »  L0:  Lt:  L,  

Now  if  the  bolts  are  of  equal  strength,  the  ten- 
sions are  proportional  to  the  deflections,  that  is 

TO :  TI:  T f      *  jo:  jt:  j8     =  L0:  Lt:  L8—  ~ 
that  is  TQ  =  C  LQ,     Tt*  C  Lt,  Q  C  T  C  : 

Hence   the   moment   about    the   back   end    holding   the 
pedestal  down,    becomes, 

C   ll   +   2  C  L2    +   2  C   L*    + C   Lj      =  SM 

Considering  now  the  gun  and  mount  together  we  have, 

K  d  -  WgLs  -  Wr  x  cos  0  =  £M 

hence    Kd  -(WSLS-  Wr  x  cos  0.) 
C  =  


L*  +  2L2  +  2L2 L2 

Oil      n 


and  the  maximum  tensiqn  to  which  the  bolt  at  the 
farther  end  is  subjected,  becomes, 


97 


[K  d  -   (WSLS-  Wr  x  cos  0)]L0 

L*  +  2  ij  +  21;  —  L; 

If  the  gun  traverses  360°  every  bolt  should  be  designed 
for  the  maximum  tension,  T0. 

The  same  method  may  be  applied  to  various  other 
combinations  for  holding  a  gun  down  on  its  foundation. 

BENDING  Itf  THE  TRAIL        In  considering  the  strength 
AND  CARRIAGE          of  a  carriage  body,  the  reactions 

at  the  trail,  Va  and  Ha,  subject 
the  total  carriage  to  a  bending 
stress. 

This  is  of  special  con- 
sideration in  field  carriages  of  the  trail  type.  The 
reaction  Va  causes  bending  while  Ha  decreases  the  bend- 
ing.  Hence  for  maximum  bending  we  should  examine  the 
conditions  for  maximum  Va  and  minimum  Ha. 
Now, 

WS(L-LS)  +  K[(L-c)sin  0  +  h^ccs  0  +  si +Pe 

•\f   s  -^— — ^_^__________^____^________________ 

a  T  : 

Ha  =  K  cos  0 
where 

L  «  horizontal  distance  between  wheel  contact  and 

spade  contact  with  ground  (in) 
c  =  horizontal  distance  from  spade  to  vertical 

plane  through  trunnions  (in) 
ht  =  height  of  trunnions  from  ground,  (in) 
s  =  distance  from  trunnion  to  line  parallel  to 

axis  of  bore  and  through  center  of  gravity 

of  recoiling  parts  (in) 
Lg  =  horizontal  distance  to  center  of  gravity  of 

total  system,  recoiling  parts  in  battery  (in) 
PS  »  powder  pressure  couple  (in/lbs) 
K  =  total  resistance  to  recoil  (Ibs) 
With  a  field  carriage,  since  the  trunnion  position 
is  very  close  to  the  wheel  contact  with  the  ground, 


98 


(L-c)sin  0  is  always  very  small  compared  with  ht  cos  0, 
hence,  we  have  approx. 

Ws(l-ls)  +  K(ht  cos  0  +  s)  +  Pe 


Va  = 


If  Lx  =  distance  from  trial  contact  with  ground  to 
any  section  in  the  carriage  body  or  trail 

hv  =  the  height  of  the  section  from  the  ground 
y 

we  have,  for  the  bending  moment  at  section  xy, 
"xy  =  Va^x  -  H  hy 

Substituting  the  value  for  Va  and  neglecting  s  being 
small,  we  have  L 

Mxy  =  [WS(L-L^)  +  K'ht  cos  0  +  Pe^"r~  ~  K  cos  0  hy 

=  WSLX(1-  ~)    +   K  cos   0    (JL  ht   -   hy)   +   Ps^i 
BENDINd  \N  TRWL  8r 


Fig.  6 


Now  from  fig. (6)  it  is  evident  y-  ht  is  always 
greater  than  hy,  hence  for  maximum  bending  moment 
we  must  have  cos  0=1,  that  is  0  =  0.    Hence  the 
maximum  bending  moment  occurs  at  horizontal  elevation, 


99 


At  horizontal  elevation,  ht  cos  0  +  s  =  h 
henca     W-(L-U)  +  K  h  +  P. 


V,  = 


a          L 
but  we  also  have  critical  stability  at  horizontal 
elevation,  that  is  K  h  +  Pe  -  WSLS  =  0  (approx.) 
therefore,  Va  =-  Ws  (approx.) 

that  is  in  virtue  of  the  mount  being  just  stable  at 
horizontal  elevation,  or  in  practice  approximately 
so,  the  vertical  reaction  at  the  spade  equals  the 
weight  of  the  entire  system,  gun  and  carriage  to- 
gether, yy  L   _  p 

Further  Ha  =  K  =   S   —          (ibs) 

and  the  bending  moment  at  section  xy  in  the  trail, 
becomes,  h 

lui    =WT      ^  W  T    -  P  ^  *          -    ( -j  n  1 K  c  ^ 

neglecting  Pe  as  usually  small  compared  with 
W$LS,  ws  have, 

Mxv  =  W_(L,  -L.  — )  (in  Ibs) 

A  jr         o     A.         K 

For  the  maximum  bending  moment  in  the  trail, 
we  consider  the  section  at  the  attachment  of  the 
trail  to  the  carriage,  then, 

Lx  =  Ls     approx.   and  therefore,  the  maximum 
B.  M.  becomes,    h  -  h 

Li       —   Iff   f      f  .^^^^^^^^^^^^  l  ill 

ro-v  \T  ~   **  c  He*   \  )  \-Li 


a  most  useful  formula  in  a  prelinary  carriage  layout 

It  is  important  to  note  that  if  the  recoil 
varies  the  above  formula  and  analysis  do  not  hold. 
When,  however,  the  recoil  varies  on  elevation  the 
maximum  bending  moment  in  the  trail  is  obtained  at 
the  minimum  elevation  where  the  short  recoil  COTD- 
mences,  that  is,  when  cos  0  is  a  maximum  for  the 
minimum  recoil. 


If  b_  =  the  short  recoil  at  maximum  elevation, 


then, 
we  have, 


KS  -  maximum  total  resistance  to  recoil,  then, 


100 


W.L^Cl  -  ~)  *  K  cos  0  (^1  ht  -  hy)  +  Pe  JL 

where  Lx  =  distance  from  trail  contact  with  ground 
to  any  distance  in  the  carriage  body  or  trail. 

h>  *  the  height  of  the  section  from  the  ground. 

Pe  3  maximum  powder  pressure  couple. 


EXTERNAL  REACTIONS  DOSING       Counter  Recoil  may 
COUNTER  RECOIL  be  divided  into  two 

periods,  the  accelerating 
and  the  retardation  period 
so  far  as  the  external 
effects  on  the  mount  are  concerned. 

During  the  accelerating  period,  the  external  re- 
actions on  the  recoiling  parts  alone,  are  the  elastic 
reaction  of  the  recuperator  in  the  direction  of  motion, 
the  guide  and  stuffing  box  frictions  and  a  hydraulic 
resistance  during  the  whole  or  part  of  the  accelerat- 
ing period,  together  with  the  component  of  the  weight 
of  the  recoiling  parts  parallel  to  the  guides,  oppos- 
ing the  notion  of  counter  recoil. 
Hence,  if 

x  =  the  displacement  from  beginning  of  counter 
recoil  of  the  recoiling  parts  with  respect 

bo  guides. 
Ffa=  the  resultant  accelerating  force  of  counter 

recoil. 
Kra  the  resultant  retarding  force  of  counter  re- 

coil. 
Fx  *  the  recuperator  reaction  for  displacement 

x  trom  beginning  of  counter  recoil. 
R  =  the  total  friction. 
Hx3  the  hydraulic  resistance,  if  any,  of  throttling 

through  recoil  orifices  or  counter  recoil  buffer. 
Then,  during  the  accelerating  period 


101 


d  x  i  i 

m  j^j-  =  Fx  -  R  -  Hx  -  Wr  sin  0  =  Ka 

and  for  the  subsequent  retardation 
.  t 

-  m  =  R  +  Hx  +  W_  sin  0  -  F  =  K' 

dt* 

Considering  now  the  external  forces  on  the  total 
system  (recoiling  parts  together  with  mount)  the 
braking  resistance  for  the  recoiling  parts  then  be- 
come internal  reactions,  and  considering  inertia  as 
an  equilibrating  force,  we  have,  as  before  the  fol- 
lowing external  forces, 

•  d*x 

Ka  =  m- — -   The  inertia  resistance  during  ac- 
celeration which  is  opposite  to 
C'recoil. 

•  »    d'x 

*r     md~T*   ^ne  inertia  resistance  during 

retardation  which  is  in  the  direction 

of  C'recoil. 

Wr  *  Wt.  of  recoiling  parts 
Wa  *  Wt.  of  carriage  proper 

Ha  and  Va  Horizontal  and  vertical  reactions  of 
spade  and  float 

y^  *  Front  Pintle  reaction  -  horizontal  com- 
ponent assumed  zero  as  before. 
During  the  accelerating  period,  obviously, 
Ka  <  Kr     that  is, 


FX-R-HX  -  Wrsin  0  <  Fx+  R  +  Hx  -  Wrsin  0 

hence,  so  far  as  stability  and  the  balancing  re- 
actions exerted  i>y  the  ground  or  platforu  on  the 
carriage  mount  are  concerned,  the  external  effect 
during  the  acceleration  period  of  counter  recoil 


102 


need  not  be  considered. 

If  now,  the  inertia  resistance  is  considered  as 
an  equilibrating  force,  we  have 

Kr  (d+L  sin  0)-Wr [ (L-Lr)+b-x  cos  0]-  Va(L-La)+VaL=  0 

Let  d   =  d  +  L  sin  0  =  ht  cos  0  +  (L-  c)  sin  0  +  S 

Hence  the  limitation  for  counter  recoil  stability, 
noting  that  Wr(L-Lr)  +  Wa(L-La)  =  Ws(L-La)  becomes 


Krd'  =  Ws  (L-LS;  +  Wr(b-x)  cos  0 


For  a  constant  marginal  counter  recoil  stability 

. 
moment  O1  this  equation  becomes 

Krd'  =[G'+WS(L-LS)  +  Wr  b  cos  0]  -  Wr  cos  0  x 
and  the  stability  slope  for  a  constant  marginal  counter 
.ecoil  stability  is  evidently 

,     Wr  cos  0 


that  is  decreasing  as  the  recoiling  masses  move  into 
battery.   Minimum  stability  is  evidently  in  battery 
position  and  0=0,  that  is 

Wf  r   r  \ 
s  \L>  -L>s  I 


h 

where  h  =  d   for  0=0 

In  ordinary  field-  carriages,  the  weight  of  the 
system  in  battery  is  very  close  to  the  wheel  axle 
or  contact  of  ground  and  wheel,  consequently  (L-LS) 
is  very  small . 

Therefore  counter  recoil  stability  is  the 
primary  limitation  in  the  design  of  a  counter  recoil 
system. 

STABILITY  The  question  of  stability 

for  field  carriages  is  of  fundament- 
al importance,  it  being  a  primary 
limitation  imposed  on  the  design 
of  a  recoil  system.   If  a  gun 

carriage  is  to  be  stable,  then 

Kd  -  WaLa  -  *r(lr  -  x  cos  #)  =  0 


103 


If  we  have  a  constant  marginal  moment  G,  that 
is  an  excess  stability,  we  have 


Kd  -  WaLa  -  Wr  (lr  -  x  cos  0)  =  G 
K  = 


-G  +  VTSLS  -  x  v»r  cos  0 


=  A  -  m  x 
A  = 


where    — G  +  W  I          W  cos  0 

\*      •  »  «  L  »  !•«»  w  W   Jv 


Thus  the  resistance  to  recoil  to  conform 
with  a  constant  margin  of  stability  decreases  in  the 
recoil  proportionally  to  the  distance  recoiled  from 
battery. 

In  battery,  the  resistance  to  recoil, 
-G  *  WSLS 

Kb  =  A  =  — 

and  put  of  battery,  the  resistance  to  recoil  becomes, 
where  b  =  total  length  of  recoil 

-G  +  WSLS     Wr  b  cos  0 

d  d 

consequently,  fot  a  constant  margin  of  stability, 
Wr  b  cos  0 


Kb 


d 

From  this  we  obtain  the  equations  of  resistance  to 
recoil  for  constant  stability  against  displacement, 

Wr  cos  IS  -G  +  WsLg 

Kx  =  Kb x,  where  Gb  = 

d 

In  our  Ordnance  Department,  KX  =  KQ  =  a  constant 

during  the  powder  pressure  period. 

Thus  if  B  represents  the  corresponding  length  of 
recoil,  then  for  a  constant  stability  moment  G, 
-G  +  WSLS    E  Wr  cos  0 

K /-\  "~      v        ~       ,      '     A  ~~  1 


104 


*r  cos  0 
and  Kx  -  K0 (x  -  E) 

Obviously  the  stability  "slope"  or  space  rate 
of  change  of  resistance  to  recoil  for  constant  margin 
of  stability,  is    ^  cos  0 

ID  =  — — — — 

d 
where 

d  =  ht  cos  0  +  s  -  c  sin  0 

A3  the  gun  elevates,  Wr  cos  0  remains  finite, 
while  d  decreases  to  zero  at  the  elevation  Si,  where 
the  line  of  action  of  the  resistance  to  recoil  passes 
through  the  spade  point. 

Thus  the  stability  slope  "m"  thereby  increases 
to  an  infinite  value  at  that  same  elevation. 

But  it  is  important  to  note  that  the  resistance 
to  recoil  out  of  battery  is  finite  and  increases  con- 
siderably as  "d"  decreases  so  far  as  it  is  limited 
by  stability. 

Obviously  in  design  it  is  inconsistent  to 
•ake  the  slope  of  the  space  rate  of  change  of  resist- 
ance to  recoil  consistent  with  the  stability  slope 
as  the  gun  elevates,  since  the  stability  becomes 
sufficiently  increased  to  allow  a  large  resistance 
to  recoil  bo  be  used. 

We  may,  therefore,  cause  the  slope  to  vary  arbitrar- 
ily as  a  linear  function  from  a  maximum  value  at  an 
arbitrary  low  angle  of  elevation,  say  some  value  from 
0°  to  6°,  to  zero  at  the  angle  of  elevation  where  the 
resistance  to  recoil  passes  through  the  spade. 

Thus  if, 

0°  =  the  initial  angle  or  lower  angle  of  elevation 
from  which  the  slope  is  to  decrease 
arbitrarily. 

£Jt  *  the  angle  of  elevation  corresponding  to 

where  the  resistance  to  recoil  passes  through 
the  spade. 
do  =  moment  arm  of  resistance  to  recoil  about 

spade  point  for  angle  to 


105 


8  SOO  0  .j*      8^8   *  *J  "  Z  g* 

L,  tf 

m  =  stability  slope  for  any  angle  of  elevation  0 

Wr  cos  j0 

m  =  -  =  stability  slope  at  lower  angle 
o     do 

of  elevation. 


m  =  m0  -  k  (0  -  00) 


then, 

m  =  m0  • 

At  angle,  of  elevation  0t,  o  =  0   hence 

•o  *  k^t  -  *o>  °r  k  =   TO°. 

0-0Q 

hence 


m  =  mo   -  (-zr\ TT— )  (#  -  00)  or  substituting  for 

i  "   o 

Wr  cos  00      0  -  00      Wr  cos  00  0t-  0 


Thus  the  variation  of  the  space  rate  of  change 
of  resistance  to  recoil  may  be  divided  into  two 

periods, 

(1)  from  0°  to  0Q 

w  ••  ~  C^™  ci 

Wr  cos  0 
OB  = which  is  parallel 

to  the  stability  slope 

(2)  from  00  to  0° 

Wr  cos  08  0-0 


, 

where  the  slope 

ln         W  ~  &  f\ 

is  arbitrary. 

A  graph  of  the  variation  of  the  space  rate  of 

change  of  the  resistance  to  recoil  against  elevation 
conforming'  to  the  assumption  (1)  and  (2). 

If  there  is  always  to  be  an  excess  stability 
couple  G  we  have  from  the  previous  discussion,  fixed 
limitations  for  the  resistance  to  recoil  in  and  out 
of  battery. 

Thus,  from  0°  to  0° 

o 


106 


-  G    +   WSLS  -  G   +   WSL3          Wr  b   cos  ft 

Kv    =    - :       k    =    : - 


where  throughout  recoil  G  is  a  constant  marginal 
stability  couple,  and  from  0O   to  0t  - 
-G  +  WSLS     »r  b  cos  0 


the  length  of  re- 
coil  being  as  be- 
fore shortened  as 

the  gun  elevates  but  if  the  stability  marginal  move- 
ment is  never  to  be  decreased  for  any  part  of  the 
recoil  below  G,  since  the  stability  slope  and  space 
rate  of  resistance  to  recoil  increase  and  decrease 
respectively  as  0  increases  from  0Q  to  0   it  is  ob- 
vious that  the  minimum  stability  is  in  the  position 
of  out  of  battery. 

Therefore  the  resistance  to  recoil  in  battery 
is  the  resistance  to  recoil  out  of  battery  with  a 
marginal  moment  G  of  actual  stability,  augmented  by 
m  b  . 

That  is, 


K   = 


-G   +  WSLS 

Wrb   cos  0 

Wrb   cos  00         ji 
+                               r 

J  -  0 
1             ] 

d 

d 

do                < 

»J=*0 

cos  0         cos 

00       0-  0 

f        *                    1    \ 

LENGTH  OF  RECOIL         Obviously  the  overturning 
CONSISTENT  WITH      force,  that  is  the  resistance  to 
STABILITY  OF  MOUNT   recoil,  is  a  function  of  the 

length  of  recoil  varying  roughly 
inversely  as  the  length  of  re-  v 

coil.   Hence  as  the  gun  elevates  the  stability  in- 
creases and  the  recoil  may  therefore  be  shortened. 

In  a  preliminary  design  it  is  desirable  to  know 
the  length  of  recoil  as  limited  by  stability,  from 
0°  or  the  lowest  elevation  wherein  stability  is  de- 
sired to  the  elevation  0°  where  the  stability  slope 


107 


is  made  to  change  arbitrarily. 

Let  Cs  =  the  constant  of  stability  = 

Overturning  moment 

= where  the  overturn- 
Stabilizing  moment 

ing  moment  =  Krd  and  the  stabilizing 
moment  =  WCL_  -  W_  b  cos  0 

OO  I 

We  may  consider  the  limiting  recoil  at  various  elevations 

(1)  with  a  constant  resistance  to  recoil 

as  would  occur  in  certain  types  of  re- 
coil systems. 

(2)  with  a  variable  resistance  to  recoil 
using  a  stability  slope  as  outlined  in  the 
previous  paragraph- 

For  a  constant  resistance  to  recoil  :  =  K  : 

The  critical  position  of  stability  is  obviously 
with  the  gun  at  the  end  of  recoil  out  of  battery. 

Then     CS(WSLS  -  Wr  b  cos  0) 
K  =  -       -r- 

>"  -sj  •        :?f.Q  -j^bwcq  arfJ  $.«iii;fe 

1  va 

2  mr  Vf 

K  = Ses  "DYNAMICS  OF  RECOIL".  Chap.  VI. 

b-E  +VfT 

Where  E  =  displacement  during  powder  period  in  free 
recoil. 

T  =  total  time  of  free  recoil. 

Vf  =  Max.  free  velocity  of  recoil, 
hence 


CS(WSLS  -  Wr  b  cos  £5) 


b-B+VfT  d 

The  above  equation  reduces  to  the  quadratic  form 

Ab2  +  Bb  +  C  =  0   and  its  solution  is, 

/—* 

-  B  ±  /  B   -  4AC 
b  =  — - 


2  A 


108 


Where  A  =  Wr  cos 


B  =  Wr  cos  0  (VfT  -  S)  -  WSLS 
C  =  WsLs(VfT  -  E) 


For  rough  estimates,  especially  where  the  length 
of  recoil  is  comparatively  long,  we  may  assume, 

^  mrVf    CS(WSLS  -  Wr  b  cos  0) 


-  B  +  /B2  -  4  AC          Cs  Wr  cos 

b  .  » and  A  .  - 


C  =  -  mr  V£ 
For  Variable  Resistance  to  Recoil: 


-  -  --  --  ----- 

The  resistance  to  recoil  is  assumed  constant 

during  the  powder  pressure  period  and  thence  to  de- 

crease uniformly  with  a  stability  slope  as  given  in 
previous  article.   Therefore,  from  the  end  of  the 

powder  pressure  period  to  the  end  of  recoil,  the 
stability  factor  remains  constant  from  0  to  0O 

(i.e.  to  where  'the  stability  slope  is  made  to  change 
arbitrarily)  . 

At  the  end  of  the  powder  period.  (See  Dynamics 
of  Recoil):  a 

Kd  =  G.(»3LS  -  Wp  (E  -  -—-)  cos  0 

2m_ 
hence 

CS(WSLS  -  *rE  cos  6) 


do   I  w 

--  COS  0 


Now  the  resistance  to  recoil  out  of  battery  at  the 
end  of  recoil,  becomes, 


109 


KT* 

K  -  m  (b  -  E  +  -  )  (See  Dynamics  of  Recoil) 
2m  r 


hence  by  the  equation  of  energy 

VT*  KT2  KT       9 

[2K-m(b-E+  £-)]    <b-E+  g-j   -  Mr(Vf   ----  )2 
/snip  «••  nr 

Expanding   and  simplifying,    we   have    the   quadratic 
form:      Ab*   +   Bb   +  C  =  0 


-  B  +   B2  -  4AC 
where  b  =  -  —  - 

H 

fr  cos  0       0 
and  A  =  m  =  Cs  —  7  -  from  0   to  0 


B  =      -  2K  -  2mB 

ID. 


c  =  [2C 2VfT)  K  + 


C(WL  -  WE  cos  0) 


From  #o  to  0t  degrees,  the  stability  slope  is 

made  to  change  arbitrarily,  decreasing  proportionally 
with  the  elevation  from  the  stability  slope  at  j0o  to 
zero  slope  at  0^  where  the  line  of  action  of  the  re- 
sistance to  recoil  passes  through  the  spade  point. 
The  critical  stability  is  obviously  at  the  end  of 
recoil,  and  the  resistance  to  recoil  in  battery  (K) 
is  the  resistance  to  recoil  out  of  battery  (k)  aug- 
mented by  the  product  of  the  length  of  recoil  from 
the  end  of  the  powder  period  to  the  end  of  recoil 
multiplied  by  the  arbitrary  stability  slope  (m). 
From  the  energy  equation,  we  have, 


110 


2 

KT  KT 

(K  +  k)  (b  -  E  +  £L->  =  m  (Vf  -  — ) 
2rar  mr 

now  K  =  k  +  m(b-E  +  — ) 
2mr 


K  =  5 —  =  the  constant  resistance  to 

mT 
1  _  recoil  during  the  powder 

2rar     period, 

and      CS(WSLS  -  Wr  b  cos  0) 

k  =  =  the  resistance  to 

d 

recoil  at  the  end 

of  recoil. 

Substituting  these  values  in  the  enery  equation, 
we  obtain  a  quadratic  equation  in  "b!!   A  sufficient 
approximation  and  simplification  can  be  made,  by  not- 
ing that    2 

E  -  — —  -  0.9  E  approximately  and 
2mr 

KT 
Vf  =  0.9  Vf  approximately 

Therefore,   (K  +~k)  (b-  0.9E)  =  0.81  mr  vj 
and  K  =  k  +  m(b-  0.9E) 

Cs(WsLg  -  Wr  b  cos  0) 

= +  m(b-0.9E) 

d 

substituting  in  the  energy  equation,  we  have, 

2Ca 

(W_LS  -  Wr  b  cos  J0)  +  m(b-  0.9E)(b-  0.9E)  =  0.81  m_Vf 

d  z 

Reducing  and  simplifying,  we  have  the  quadratic  sol- 
ution,        ,— 

-B  ±  /  B   -  4AC 
b  =  — 


2A 


s 
where  A  =  m  -  —7-  Wr  cos  0 

2C, 


III 
2Cf 


*  °-9E  wr  cos 


2 

Q  =  0.8l(mE  -  mrVf 


o 
from  0   to 


Wr  cos  00   0t  -  0 

m  ,  (      )   from  0   to  0 


For  a  close  approximation  and  when  the  resist- 
ance to  recoil  is  not  constant  during  the  powder 
period,  if 

K  =  the  resistance  to  recoil  in  battery 
k  =  the  resistance  to  recoil  out  of  battery, 
we  have, 

Kl-     fe  m     V  ? 

r        R  111  M  V    f 

( >b    = (approximately) 

but   K  =  k  +  mb 

cs 

and  k  =  7-  0»_L,   -  W_b   cos  0) 

d  as 

Substituting,   we   have 

20  s 
[-•—  (WSLS   -  Wr  b    cos   0)    +   mb]   b   =  mrVf 

and   the   value   b,    becomes, 


-B  ±  /V  -  4AC 
b  = 


2A 

where  pp 

3   nr 

A   =   m  -   ~r~*'r     cos   « 
.fee  d 

2CS 

B  =  -r  WSLS 


c  =  -  «rvf8 

N 

Wr  cos  0 

ra  =     " or  any  arbitrary  slope  as  desired. 
d 

The  above  formula  is  sufficiently  exact  for  a 
preliminary  layout  with  a  variable  recoil  and  resist- 
ance to  recoil  provided  the  margin  of  stability  is 
chosen  fairly  large,  that  is  when  a  low  factor  of 
stability  is  taken. 

JUMP  OF  A  FIELD  CARRIAGE       When  the  overturning 

moment  exceeds  the  stabiliz- 
ing moment,  we  have  unstabil- 
ity  and  an  induced  angular 
rotation  about  the  spade 

point.   After  the  recoil  period,  the  gun  carriage  is 
returned  to  the  ground  by  the  moment  of  the  weights 
of  the  system.   This  phenomena  is  known  as  the 
jump  of  the  carriage. 

For  the  condition  of  unstability,  we  have: 

K  d  ~  *s^s  *  *r  cos  >  ° 
where  as  before, 

K  =  total  resistance  to  recoil 

Wg=  weight  of  entire  gun  carriage  including  gun 
13=  distance  from  spade  contact  with  ground  to 

center  of  gravity  of  total  system  in  the 
battery  position. 

wr=  weight  of  the  recoiling  parts 

x  =  movement  in  the  recoil  of  the  gun. 
To  analyse  the  motion  of  the  system,  consider 

(a)  the  recoil  or  accelerating  period. 

(b)  the  retardation  or  return  period. 

The  recoil  period  may  be  subdivided  into  the 


113 


powder  period  and  the  pure  recoil  period.   During  the 
recoil  period  the  gun  and  gun  carriage  are  given  an 
angular  velocity  which  reaches  its  maximum  at  the 
end  of  recoil.   During  the  retardation  the  angular 
velocity  is  gradually  decreased  to  zero,  but  with 
increased  angular  displacement,  the  maximum  angular 
displacement  occuring  when  the  angular  velocity 
reaches  its  zero  value.   Further  change  in  angular 
velocity  results  in  a  negative  velocity  and  a  corres- 
ponding angular   return  of  the  mount  to  its  initial 

position. 

ttniliootn  jo  x-*Y"''^ 
The  acceleration  during  the  recoil  period  is  not 

constant,  even  with  constant  resistance  to  recoil, 
due  to  the  fact  that  the  moment  of  inertia  and  the 
moment  of  the  weights  of  the  recoiling  parts  about 
the  spade  point  varies  in  the  relative  recoil  of  the 
gun.   Therefore,  the  angular  acceleration  is  not 
constant  during  the  accelerating  period.   Likewise 
during  the  return  of  the  recoiling  parts  into 
battery.   Further  the  effect  of  the  relative  counter 
recoil  modifies  the  return  angular  motion. 

Consider  the  reaction  and  configuration  of  t"he 
recoiling  parts  and  carriage  mount  respectively. 
See  figure  (7). 


114 


Let  X  and  Y  =  the  components  of  the  reaction  between 
the  recoiling  parts  and  carriage  mount, 

parallel  and  normal  to  the  guides  res- 
pectively. 

M  =  the  couple  exerted  between  same. 
Ia=  the  moment  of  inertia  of  carriage  mount 

about  the  spade  point. 

Ir=  moment  of  inertia  about  the  center  of  gravity 
of  the  recoiling  parts. 

dx  =  perpendicular  distance  from  spade  point  to 
*  i 

line  of  action  of  X. 

dx  =  perpendicular  distance  from  X  to  center  of 

gravity  of  recoiling  parts. 

d  =  dx  +  dx  =  perpendicular  distance  to  line 
parallel  to  guides  and  through 
center  of  gravity  of  recoiling  parts  from 
the  spade  constant  with  ground. 
Q  =  angle  made  by  d  with  the  vertical 
0  =  angle  of  elevation  of  the  gun  (in  battery) 
x  =  distance  recoiled  by  gun  from  battery  position 
x0s  distance  from  "d"  to  center  of  gravity  of  re- 
coiling parts  in  battery  measured  in 
direction  of  X  axis  of  perpendicular  to  line 
d. 
r  =  distance  from  spade  point  to  center  of  gravity 

5         J 

of  recoiling  parts, 
e  =  angle  r  makes  with  vertical 
lr=  horizontal  distance  to  center  of  gravity  of 

recoiling  parts  from  spade  contact  with 

ground . 
Wa  =  weight  of  carriage  proper  (not  including 

recoiling  weights) 
ra  =  distance  from  spade  point  to  center  of  gravity 

of  carriage  proper, 
o  =  angle  ra  makes  with  horizontal 
la  =  horizontal  distance  from  spade  point  to 

center  of  gravity  of  carriage  proper. 
Then 

lr  =  (xQ-x)  cos  6  -  d  sin  6 


115 


la  =  ra  cos  (9  +  a  -  0) 

where  in  battery  6  =  0  and  for  any  other  angular 
position  during  the  jump  of  the  carriage, 

6=0+6       B=a  variable  angle  during  the  jump, 
For  the  angular  motion  about  the  spade  point, 

For  the  carriage  mount,  without  the  recoiling 
parts, 


d  6 
i    "  "u  a~*   "**  d  +  2 


Xdx  -  Y(x0  -  x)+  m  -  wala  =  Ia  T— -    (i) 


and  for  the  recoiling  parts, 

adding  (1)  and  (2),  we  have, 

d*6       (3) 
Xd  -  Y(xQ-x)-  wala  =  (Ia  +  Ip)  — - 

Since  the  recoiling  parts  are  constrained  to 

rotate  with  the  carriage  mount,  they  partake  an 
angular  acceleration  about  the  spade  point  combined 
with  a  relative  acceleration  along  the  guides. 

The  acceleration  of  tne  recoiling  parts  is 

AfT   Jtfc>      9& 
divided  into: 

(1)     The  tangential  acceleration  of  the 
recoiling  parts  about  the  spade  point; 
due  to  the  constraint  in  the  guides, 


dt2    a°d  is  divided  into  components  in  the 
x  and  y  direction 

d*e    /«      j  d2e    ) 

-  cos  (e  +  e)  =  d  —  -—    f 
dta  dt2 


116 


(2)     The  centripetal  acceleration  of 
the  recoiling  parts  about  the  spade 
point  due  to  the  constraint  in  the 
guides, 

s    ,de  .  2 
rw  -  r  (.    ) 

dt      and  divided  into  components  in 

the  x  and  y  direction. 

,d9,*     ,  N,d6v! 

r(— )   sin  (9  +  e)  =  (x0  -  x)(— 4 


r(- — )*  cos  (e  +  e)  =  d 
at 


(3)  The  relative  acceleration  of  the 
recoiling  parts 

d'x   <*vr 

——  —  -  -T  —  along  the  x  axis 

(4)  The  relative  complimentary  centri- 
petal acceleration  due  to  the  combined 
angular  and  relative  motion  of  the  re- 
coiling parts: 

de 


(5)     The  angular  acceleration  of  the 
recoiling  parts  which  obviously  equals 
the  angular  acceleration  about  the 
spade  point,  that  is 

d2e 


dt! 


For  the  motion  of  the  recoiling  parts 
along  the  x  axis,  we  have 

dvr       d% 
Pb  -  m_ -  ard  4  wr  sin  e  -  mp(xQ- 

dt         dt« 
0  (4) 


117 


For  the  motion  of  the  recoiling  parts 
normal  to  the  guides, 

x  d*e  de    ,,de  .a 

Y  -  ra_(xn-x)  --  wr  cos  9+2  nrvr  -  +  m_d(—  )  = 
dt»  dt       d  t 

0  (5) 


Substituting  (4-)  and  (5)  in  (3)  we  have, 

1  2 

2       2 


ur  1  —  mi  1   +  9m  v  ( v  —  v  ^ 

-   -  aL    .      _     at_    "r^r  Wa1a   60Brv,Axo  x; 

*9         d29 

(Ia+Ir) — -  =  0        (6) 

dt         dt 
where 

Ir=(x0~x)  cose  -  d  sin  0  ) 

'  functions  of  the 

,  .  variable  angle 

la  =ra  cos  (e+a-6)        )  e 

From  equations  (4)  and  (6),  we  have  8  as  a 
function  of  t.   An  exact,  solution  of  these  dif- 
ferential equations  is  complicated  and  therefore  an 
approximate  solution  must  be  resorted  to. 

APPROXIMATE  SOLUTION  09  THE  JUMP  Of  A  FIELD 
CARRIAGE. 


The  static  equation  of  recoil,  that  is  the 
equation  of  motion  of  the  recoiling  parts  upon  the 
carriage  is  stationary,  becomes, 
s 

>•  W_  sin  e  -XR  *  0 


and  the  equation  of  motion  of  the  recoiling  parts 
along  the  guides  when  the  carriage  jumps,  becomes, 

9  -  X  -  o,rd  L£  -«r  <x0-x>  (A  0 


118 


Now  the  term  mr(xrt-x)( — )   is  small  and  may  be  neg- 

dt 
lected,  but  on  the      ,2 

d  9 
other  hand  the  term  m_d  may  be  considerable. 

dt2 
Furthermore  the 

braking  X  and  Xs  may  differ  considerably  as  well. 
The  term    2 

Pb-mr  - — -=  Ks   in  static  recoil. 

whereas  with  the  jump  of  a  carriage 

,2 

Pb-  mr  =  cKs  where  c  =  0.9  approx. 

dta 

During  the  pure  recoil  on  retardation  period  of 
the  recoiling  parts,  we  have 

m_- — -  =  K,     in  static  recoil. 
rdt2          

whereas  when  the  carriage  jumps, 

,2 

m_  — -  =  cK,  where  c  =  0.7  to  0.9 
r  dt8 

Considering  the  moment  equation  for  the  movement  of 
the  total   mount  about  the  spade  point,  we  have, 

(pb-rar  — *  )d-[mr(d2+(x0-x)2)+Ir+Ia] +2mrvr(xo-x) 

dt2  dt2 

d6 

_   **rlr   _   „   I     _   Q 

dt          a  a 


where  lr=(xo-x)  cos  6  -  d  sin  6 

la=r_  cos  (0  +  a  -v  ) 

de 
The  term  2mrvr(xo-x) —  is  always  small  and  may  be 

Q  t 

neglected . 

If  b  =  length  of  recoil,  for  the  average  during 
the  recoil, 

let 

b 

v  — y  —  Y  —  •«- 

*o  x    °   2 


119 


Then,  we  have,  for  an  approximate  solution, 

2 

cKd-[mr(d%(x0-  |)2  +  Ir  +  Ia]  i-i  -  wsls  =  0    (7) 


where  wsls  =  «ala   +  «rlr     c  =  0.8  to  0.9 

and  ws  =  wa  +  wr   and  approximately,  if  the  jump  is 
small, 

lr=  (x0-  -)  cos  0  -  d  sin# 

_  .        ~  _  .  ..    i  '  t 

la=  ra  cos  a 

0  =  angle  of  elevation  of  the  gun 

b  =  length  of  recoil 

d  =  perpendicular  distance  of  line  parallel 

to  guides  and  through  center  of  gravity 

of  recoiling  parts  from  spade  contact  with 

ground. 
XQ=  the  perpendicular  distance  from  d  to  the 

center  of  gravity  of  the  recoiling  parts 
ra  =  distance  from  spade  point  to  center  of 

gravity  of  carriage  proper 
a  =  angle  ra  makes  with  horizontal 
Hence  for  the  angular  acceleration, 

d29    _  eKd-  wsls  _        rad 


,2          x,2         D72 I I  - 

dt    mr(d  +xo-  -)  +Ir+  Ia  sec* 

If  we  assume  a  constant  acceleration,  we  have, 

for  the  angular  velocity  attained  at  the  end  of  recoil, 

d0          (cKd  -  w,,ls)t  rad 

( \  

At  2   1 


mr(d(x0-  |)-Ir-Ia 
where  t^  =  the  time  of  recoil   we  have  approximately, 


cKs 

wv  -H  w  4700 
where  V  =  0.9       


120 


w=  weight  of  projectile   (Ibs) 
w=  weight  of  charge       (Ibs) 
ws=  weight  of  recoiling  parts   (Ibs) 
c  s  0.9  approx.  and  tp=  the  total  powder  period  ob- 

tained by  the  methods  of 
interior  ballistics. 

The  angular  displacement  during  the  first  period 
of  the  jump,  becomes, 

t   (cKd-wsl3)t* 

6  =  -  B  radius. 

- 


During  the  second  period  of  the  jump  we  have, 
the  angular  velocity  decreasing  but  the  angular 
displacement  still  increasing:  then 


rad 

~ 


Integrating,  we  have 


t     «r(d»(x0-  |)2)+Ir+Ia 
and  for  the  angular  displacement, 


SI  *,!<  (-  -Ox*  L)^m    it 
-  ws!3  t de 

-  *)2)  +1,  +1,  +  ~  fc  "  9l 


To  determine  the  time  of  jump  required  to  attain 
the  maximum  angular  displacement,  we  have  the 
angular  velocity  reduced  to  zero,  whence, 

aet      wsis  ta 

»  j  i.  '  ~  , 

from  which  we  may  determine  ts>  Therefore,  the 


121 


maximum  angular  displacement,  becomes, 

_I *rde 

mr^d2+(x0-  ' 
2 


tf 


The  effect  of  counter  recoil  is  to  increase  t 

2 

and  decrease  the  negative  moment  (-  w0l  ). 

S  S 


RECAPITULATION  OF  FORMULAS: 


EXTERNAL  EFFECTS  AND  STABILITY, 


Resistance  to  recoil: 

J  Mf 

K  =  (Ibs)  constant  resistance  through- 

b-E+Vfr 

out  recoil. 

mp=  mass  of  recoiling  parts  =  wr   (16s) 

I 

g  =  32.16  ft/sec. a 
b  =  length  of  recoil  (ft) 
E  =  free  recoil  displacement  during  powder  period 

(ft) 

T  =  time  of  free  recoil   (sec) 
Vf=  Max.  velocity  of  free  recoil   ft/sec. 

BETHEL'S  FORMULA 

.  mrvf         1 

(Ibs)  Constant 

lD  +  (.096  +.  0003d  )M  Vf   resistance. 

Vo 

M  =  travel  up  bore  (inches) 
VQ=  muzzle  velocity  (ft/sec) 

d  =  diam.  of  bore    (inches) 


Assuming  a  gun  carriage  to  be  supported  by  a 
hinge  joint  at  the  rear  (A)  and  a  vertical  support 


122 

in  the  front  (B)  we  have  the  following  equations  for 

the  reactions  of  the  supports: 
Let 


Ha-=  horizontal  component  at  rear  hinge  support 

or  spade  of  carriage.  (Ibs) 

Va=  vertical  component  at  rear  hinge  support  (Ibs) 

Vjj=  reaction  of  front  support  assumed  vertical 

(Ibs) 
L  =  horizontal  distance  between  carriage  supports 

(in) 
ht=  height  of  trunnion  above  support        (in) 

s  =  perpendicular  distance  from  center  of 

gravity  to  recoiling  parts  to  line  of  action 
of  the  resistance  to  recoil  (in) 

c  =  horizontal  distance  from  rear  support  to 

trunnion  (in) 

K  =  total  resistance  to  recoil  (Ibs) 

<&  =  angle  of  elevation  of  gun 

g  =  vertical  distance  from  ground  to  horizontal 
.component  of  resultant  spade  reaction. 

IN  BATTERY:    For  low  angles  of  elevation: 

w<.L»-K(h+  cos  0  +  s-c  sin  0)-Pe          ' 

:v  -  '  - 


K[ht+  cos  0+(L-c)sin 


)  Ha  =  K  cos  0 


123 


For  high  angles  of  elevation 

wsL,  +K(c  sin  0  -ht  cos  0  -  s)-  Pe         ) 

)  Vb  =  ( 

ws(L-Ls)  +K[(L-c)  sin  0  +  h  +  cos  0+s]+Pe   <  (Ibs) 

(  va  -         -r  ~) 

(  Ha  =  K  cos  0  ) 

OUT  OF  BATTERY:    For  low  angles  of  elevation: 

w"3Ls  -  Wrb  cos  0-K(ht  cos  0  +  s  -c  sin  0 


(  WS(L-LS)  +W_b  cos  0  +K(b+  cos  0+(L-c)sin0+a> 

)  Va  =  -  *—  -  ( 

)  ( 

(  Ha  =  K  cos  0                                 ) 

For  high  angles  of  elevation: 

(  ) 

)  wgLs-»rb  cos  0+K(c  sin  0  -ht  cos  0-s)      ( 

(  Vb  •  -  L  -     ) 

)  ( 

(  »?g(L-Ls)  +»rb  cos  0  +K(htcos  0(L-c)sin0+s)  ) 


Ha  »  K  cos  0  ( 

With  a  field  carriage  where  the  spade  is  in- 
serted in  the  ground,  the  center  of  pressure  lies  a 
distance  "g"  inches  vertically  down.   The  general 
equations  for  the  support  of  a  field  carriage, 
therefore  become, 


124 

For  low  angles  of  elevation: 


wsLs-  wrx  cos  0-K(d+g  cos  0)-  Fe 

(  b  L  ) 

ws(L-Ls)+wr  x  cos  0+K(L  sin  0-d+g  cos0)+Fe 
(  Va= )• 

(  Ha  =  K  cos  0  ) 

(  d  =  ht  cos  0  +  s  -  c  sin  0  ) 

For  high  angles  of  elevation: 


\        -   *sLs     wr   x   cos   0  *K(d-g  cos   0)-Fe 
;    V^_ 


(  Va=w(L-Ls)+wr  x  coS0+K(L  sin0  -d+^  cos  0)+Fe  ) 

)  ( 

(  Ha=  K  cos  0;   d=c  sin  0-htcos  0  -  s          ) 

In  certain  types  of  Barbette  mounts,  we  have 
the  bottom  carriage  held  down  by  tension  bolts  to  a 
circular  base  plate.   If  we  draw  a  series  of 
parallel  chords  through  the  bolts  on  either  side 
of  the  axis  of  the  gun,  and  if  we  let  the  distance 
from  those  several  chords  measured  from  the  rear 
bolt,  be  LQ,  L  -  —  -  -  I»n.  we  have,  for  the  maximum 
tension  induced  in  a  tension  bolt  given  by  the  ex- 


[Kd-(wsL3-Wr  x  cos  0)JL0 

TO  =  — — : — : 


BEHDINa  III  THB  TRAIL  AND  CARRIAflB. 

Considering  the  section  at  the  attachment  of 
the  trail  to  the  carriage,  for  a  constant  length  of 


125 


recoil  the  maximum  bending  in  the  trail  occurs  at 
horizontal  elevation  and  is  given  by  the  following 
expression:          h_h 


B.K.  at  the  attachment 
of  trail  to  carriage. 

h  =  the  "height  of  the  center  of  gravity  of  the 
recoiling  parts  (axis  of  bore  practically  above  the 
ground  when  the  gun  is  in  its  horizontal  position. 

hvs=  the  height  of  the  neutral  axis  of  the 
section  above  the  ground. 

ws-  weight  of  entire  mount  including  the  gun. 

Ls=  horizontal  distance  from  the  spade  to  the 

center  of  gravity  of  the  weight  of  the  entire 

mount. 

When  the  recoil  varies  on  elevation,  the  maximum 
"bending  moment  in  the  trail  is  obtained  at  the  minimum 
elevation  where  the  short  recoil  commences,  we  have, 

^s  kx  Lx 

Mxv=wsLx(l )+Ks   cos  0S( —  ht-hv)+Pe  — - 

L  L  !-• 

where 

Ks  =  maximum  total  resistance  to  recoil  corres- 
ponding to  short  recoil  "bs. 

0S=  minimum  angle  of  short  recoil. 

Ls=  distance  from  trail  contact  with  ground  to 
any  distance  in  the  carriage  body  of  trail. 

hy=  the  height  of  the  neutral  axis  of  the  sect- 
ion from  the  ground. 

Pe  =  maximum  powder  pressure  couple. 

STABILITY  OP  COUNTER  RECOIL. 


In  the  design  of  a  field  carriage  counter  re- 
coil stability  is  a  basic  limitation.  We  have  for 
counter  recoil  stability  that. 


126 


The  equation  stability,  gives,  for  variable 
resistance  to  recoil,  for  low  angles  of  elevation 
consistent  with  the  stability  slope, 

-8+  /B*  -4AC 
b  = 


2  A 
where 


ff_cos  0  o    ^o 

A  =  •  =  Cs  -*— (from  0  to  0O  elevation) 


B»  SSI  -  2K-2mE 
mr 

and 

CS(WSLS-  wr  E  cos  0  )     (Ibs) 

w  T2 
d  -  C3   — E- —  cos  J25 

2rar 

After  an  arbitrary  elevation  00  (approx.5  )  the 
stability  of  the  mount  greatly  increases  with 
elevations  and  therefore  the  stability  slope  is  made 
to  arbitrarily  decrease  with  the  elevation  arriving 
at  constant  resistance  to  recoil  at  the  elevation 
corresponding  to  where  the  line  of  action  of  the 
resistance  to  recoil  passes  through  the  spade  point. 
To  estimate  the  minimum  recoil  allowable  for  the 
various  angles  of  elevation  in  this  range,  we  have 


-B±  /B2-4AC 
b   = 


2A        .    jsstaas   jjaiJ    BC 

0     J0.-0 

from<.00      to 


3   *  ^ 

A    =    D — -    5f_    COS    0 

d 

2Ca 
B   =   (waLs*   P.9E  wr   cos  0)-1.8  mE 


0.81 


127 


<  •  (L-t.) 

Rr  "  "~"h"        where  Kr=  the  total  resistance 

of  counter  recoil  at 
horizontal  elevation. 
ws=  weight  of  entire  mount  including  gun. 

L3=  horizontal  distance  from  spade  to  center  of 

gravity  of  ws. 
L  =  horizontal  distance  from  spade  to  wheel 

contact  with  ground. 

Further  2 

„       d  x 


(4—  f-  may  be  obtained  from  the  velocity 
curve  of  counter  recoil  towards 
the  battery  position). 
and  Kr=  Hx  +R+wr  sin  0  -  Fx  where  Hx  =  hydraulic  or 

buffer  brak- 
ing at  end  of  counter  recoil.     (Ibs) 

R  =  total  friction  resistance 

wr  sin  0  =0  weight  compound  equals  zero  at 

horizontal  elevation. 
Fx=  recuperator  reaction.  (Ibs) 

RECOIL  STABILITY 


The  stability  limitation  of  the  resistance  to 
recoil  varies  in  the  recoil  due  to  the  movement  of 
the  recoiling  weights.  The  slope  or  rate  of  the 
variation  in  the  recoil  of  the  equivalent  force 
applied  through  the  center  of  gravity  of  the  re- 
coiling parts  and  parallel  to  the  guides  that  will 
just  overturn  the  mount,  is  given  by  the  following 
expression: 

wr  cos  0 
ra= from  0°  to  0 


128 


where 

ra  =  the  stability  slope 
0  3  angle  of  elevation 

d  =  perpendicular  distance  from  spade  to  line 
through  center  of  gravity  recoiling  parts 
parallel  to  the  guides. 
wr=»  weight  of  recoiling  parts 

00=  the  initial  angle  or  lower  angle  of  elevation 
from  which  the  slope  is  to  decrease 
arbitrarily. 

If  from  0O  the  slope  is  made  to  decrease  ar- 
bitrarily with  the  elevation,  to  the  elevation  0, 
the  angle  of  elevation  corresponding  to  where  the 
line  through  the  center  of  gravity  of  the  recoiling 
parts  parallel  to  the  guides  passes  through  the  spade 
point,  we  have  for  the  stability  slope 

wr  cos  00  0t-  0 

m  = (— — ~r— )  where  the  slope  is 

d      0  ~0 

arbitrary. 

LBMQTH  Of  RECOIL  COM3ISTBNT  WITH  STABILITY 
OP  MOUNTS. 


The  equation  of  stability,  gives,  for  constant 
resistance  to  recoil, 


-mrVf    _Cs(wgLs  -wrb  cos 


The  solution  of  this  quadratic  equation  for  b,  gives: 


-B±  /B  -4AC 

b  =  

2A 


where  A=   i»r  cos  0  )   where   all   units 

8=   wr  cos  0(VfT-£)-wsLs        (   are   in   feet 
..2s  )    and   pounds. 

_  i"r»i        ci 

) 


CHAPTER        IV 


-soo 

INTERNAL   REACTIONS. 


<:  -   '       ' 

In  the  design  of  the  various  parts  of  a  gun 
carriage  it  is  of  fundamental  importance  that  we 
have  a  coraplste  knowledge  of  the  stresses  to  which 
each  member  is  subjected,  and  the  variations  of  such 
throughout  recoil  and  the  position  of  elevation  and 
traverse  . 

We  have  already  considered  the  external  reactions 
on  the  whole  system,  and  such  reactions  are  useful  in 
computing  the  stresses  in  the  supporting  structure 
for  a  gun  mount  as  the  strength  of  concrete  emplace- 
ments for  barbette  mounts,  or  the  strength  of  a  rail- 
way car  or  caterpillar  frame. 

The  primary  internal  reactions  within  a  gun  and 
its  mount  may  be  classified  as  follows: 

(a)     The  mutual  reactions  between  the  recoil- 
ing parts  and  the  carriage  proper  or  gun 
mount. 

(b  )     The  mutual  reaction,  between  the  tipping 
parts  or  cradle  and  the  top  carriage. 

(c)     The  mutual  reaction  between  the  top 
carriage  and  bottom  carriage. 

The  mutual  reaction  (a)  is  between  the  moveable 
and  statipnary  part  of  th3  total  system  during"  the 
recoil;  that  of  (b)  between  the  moveable  and  station- 

ary parts  during  elevation  of  the  gun;  and  that  of 
(c)  between  the  moveable  and  stationary  parts  in 
traversing  the  gun. 

The  mutual  reaction  (a)  may  be  subdivided  into 
individual  or  component  reactions  as  follows: 

(.1)     The  reactions  of  the  constraints 
due  to  the  guides  or  clip  reactions  at 


129 


130 


the  two  ends  of  the  clips  in  contact 
with  the  guides,  which  may  be  subdiv- 
ided into  friction  and  normal  com- 
ponents . 

(2)     The  mutual  reaction  of  the  elastic 
medium  connecting  the  recoiling  parts 
to  the  carriage  proper,  that  is,  the 
hydraulic  brake  and  recuperator  re- 
action, together  with  the  joint 
frictions.   This  will  be  known  as  the 
elastic  reaction  between  the  recoil- 
ing parts  and  carriage  proper. 
The  mutual  reaction  (b)  may  be  subdivided  into: 

(1)  The  trunnion  reaction  between  the 

tipping  parts  and  top  carriage. 

(2)  The  elevating  arc  reaction  between 
the  elevating  arc  of  the  tipping  parts, 
and  the  pinion  of  the  top  carriage. 

The  mutual  reaction  (c)  may  be  subdivided  into: 

,(1)     The  pintle  or  pivot  reaction  between 
the  pintle  bearing  on  the  bottom  car- 
riage or  platform  mount  and  the  pintle 
of  the  top  carriage  fitting  within  this 
bearing. 
(2)     The  traversing  arc  reaction,  that 

is,  the  reaction  between  the  traversing 
arc  of  the  top  and  bottom  carriage. 
These  are  usually  roller*  reactions  for 
platform  or  pedestal  mounts,  the  rol- 
lers being  either  a  part  of  the  top  or 
bottom  carriage  or  else  clip  reactions 

field  carriage  and  may  be  more 
or  less  distributed  about  the  arc  of 
contact. 

Let  X  and  Y  -  the  coordinates  of  the  center  of 
gravity  of  the  recoiling  parts  along  end  perpendicular 
to  the  guides  with  origin  at  center  of  gravity  of  re- 
coiling parts. 


131 


xt  and  yt  =   coordinates  of  front  clip  reaction 
measured  from  the  center  of  gravity 
of  the  recoiling  parts. 
&t  =  Normal  component  to  guides  of  front  clip 

reaction. 
uQt  =  Frictional  tangentional  component  of  front 

clip  reaction. 
x   and  y  =  coordinates  of  rear  clip  reactioa 

2          2  ^^^ 

measured  from  the  center  of  gravity 
of  the  recoiling  parts. 
Q  =  Rear  clip  reaction  normal  component. 

u&2=  Rear  clip  reaction  frictional  component. 
B  =  nQ  +  nQ  =  total  guide  friction. 

1         2 

B  =  elastic  reaction  (hydraulic  breaking  and  re- 
cuperator reaction  including  friction  of 
joints)  assumed  parallel  to  the  guides. 

F  =  the  total  powder  pressure  on  the  breech  of 

the  gun  which  necessarily  lies  along  the  axis 
of  the  bore. 

e  =  the  perpendicular  distance  from  center  of 

gravity  of  recoiling  parts  to  line  of  action 
F,  that  is  to  axis  of  bore. 

Assuming  as  in  Chapter  III,  the  mount  to  be 

hinged  at  the  rear  or  breech  end  to  its  support  and 
resting  on  a  smooth  surface  at  the  front  end,  and  if 
d  =  perpendicular  distance  from  hinge  to  line 

through  center  of  gravity  of  recoiling  parts, 
parallel  to  guides. 
djj=  perpendicular  distance  from  hinge  to  line  of 

action  of  B. 
lr=  horizontal  distance  from  hinge  to  center  of 

gravity  of  recoiling  parts  in  battery. 
la=  horizontal  distance  from  hinge  to  center  of 

gravity  of  stationary  parts  of  system  (includes 
stationary  parts  of  tipping  parts). 
From  fig.(l)  considering  the  reactions  on  the  re- 
coiling parts  alone,  we  have  from  the  equations  of 
motion: 


132 


Fig.  1 


133 


for  notion  along  the  x  axis,  2 

.F"—  B  —  uQ   -  uQ  +  w*r  sin  0  =  mr— —   (1) 

and  since  there  is  no  motion  along  the  .y  axis, 
.-••;  -ie*  adi  oj  ;  .-   'r^  i»fljft?j';^  i'-'ic'~ 

Qf-  Qi  =  wr  cos  ^  (2) 

tdJ  *c 

and  taking  movements  about  the  center  of  gravity  since 
there  is  no  angular  acceleration 

B(d-dh)-u(Q  y  -Q  y  )+Fe-Q  x  -  Q  x  =0   (3) 

u  22  1122 

Now   in   fig.(l)   considering    the  gun  carriage   or 
mount   including    the   stationary   parts   of   the   tipping 
parts,    we   have   for    the   moments   of    the   reaction  of   the 
recoiling   parts   on   the  gun   mount   about   the   hinge   A, 

lr-x  cos  0                                   1   -  x   cos   0 
Qt(xt* +   d   tan  0)-Q2( +d   tan  0-x    ) 

oo:     C°3  0     WxiEWUAiU^aaaal   cos  *       ?bi?ib 
+uQ    (d+y    )+uQ    (d-y    )+Bdh=   2M_Q  (4) 

1     "  1       2       2       D       ~  a 

but  uQiyi-uQzya=B(d-db)+Pe-Qixt-a8x2 
and  uQ   <-uQ  =R 

1       2 

Substituting  these  values  in  (4),  we  have, 

lr-x  cos  0 

(Qt-Q2)( +  d  tan  0)Bd+Rd+Ae=SMra 

cos  0 

that  is,  -Wr(lr-x  cos  j0)-Wr  sin  0  d+Bd+Fe+Rd=»2Mr 
or  simplifying  and  combining,  we  have, 

(B+R-«lr  sin  0)d+Fe-Wr(lr-x  cos  0)=2Mr    (5) 
at  maximum  powder  pressure,  x  is  usually  negligible 
and  the  equation  reduces  to: 

(B+R-Wr  sin  0)d+F9-Wrlr=  SMra  (51) 

From  this  we  observe  that  the  reaction  between  the 
recoiling  parts  and  the  mount  is  equivalent  in  effect 
to  a  force  (B+R-Wrsin  0),  the  line  of  action  of  which 
is  parallel  to  the  axis  of  the  bore  or  guides  and 


134 


passes  through  the  center  of  gravity  of  the  recoiling 

parts,  and  a  couple  of  magnitude  Fe,  due  to  the  powder 
pressure,  together  with  a  component  equal  to  the  weight 
of  the  recoiling  parts  and  in  its  line  of  action  assumed 
concentrated . 


Thus  the  reaction  on  the  gun  mount  of  the  recoil- 
ing parts,  therefore,  is  equivalent  to  a  single  con- 
centrated force,  the  resultant  of  (3+R-Wr  sin  0),  equal 
to  the  total  resistance  to  recoil  and  a  force  equal 
to  the  weight  of  the  system  together  with  a  couple  Fe. 
Since  a  couple  and  a  single  force  in  the  same  plane 
are  equivalent  in  effect  to  a  single  force,  parallel 
to  the  former,  and  displaced  from  it  equal  to  the 
couple  divided  by  the  force,  the  resultant  reaction 
on  the  mount  of  the  recoiling  parts  reduces  to  a  single 
force;  the  resultant  of  B+R-W  sin  0  and  Wr,  which  be- 
comes, since  B+E-Wr  sin  0  =  K,  equal  to 


J=  /K2+yf*  -2KWr  sin  0 

and  the  line  of  action  of  J  makes  an  angle 

-l(Wrcos  0)       -KW-cos  0) 
&  =Tan  *- =Tan  * 


(K+Wrsin  0)        (B+R) 

with  the  axis  of  the  bore  and  is  displaced  a  dis- 
tance Fe.  frora  the  center  of  gravity  of  the  recoiling 
J 

parts.   It  is  however,  more  convenient  in  computation 

to  resolve  this  resultant  into  its  components,  K  and 
Wr  together  with  Fe. 

If  now  we  consider  the  equilibrium  of  the  gun 
carriage  mount,  we  have  for  moments  about  the  hinge 
point,   2Mra-  WaLa  +Vb  1=0   that  is, 

(B+R-Vfrsin  0)d+Fe-Wrlr  -WaLa+Vb  1  =  C     (6) 
and  since  Wala  +  Wrir  =  WS18 


135 


136 


Equation  (6)  reduces  to 

(8+R-Wrsin  0)d+Fe-WsLs+Vb  1  =0        (6') 

which  of  course  is  the  same  as  the  equation  obtained 

in  Chapter  III. 

It  is,  however,  very  often  more  convenient  to 
regard  the  mutual  reaction  between  the  recoiling 
parts  and  carriage  as  divided  into  component  .re- 
actions along  and  normal  to  the  axis  of  the  bore  to- 
gether with  a  couple.  See  fig.  (3). 

Let  x  and  y  be  the  coordinate  axes  along  and 
normal  to  the  axis  of  the  bore  or  guides. 

Xr=  the  sum  of  the  component  reactions  along  the 

x  axis. 
Yr=  the  sum  of  the  component  reactions  along  the 

y  axis. 

Now  by  introducing  a  couple  Mr  between  the  re- 
coiling parts  and  carriage,  it  is  entirely  immaterial 
where  we  assume  the  line  of  action  of  Xr  and  Yr. 

Let  r  =  the  perpendicular  distance  from  Xr  to  the 

center  of  gravity  of  the  recoiling  mass. 
z  =  the  perpendicular  distance  from  Yr  to 

the  center  of  gravity  of  the  recoiling 
mass. 

Considering  the  equations  of  motion  of  the  re- 
coiling mass,  we  have, 
,2 
P*wrsin  /HCr=Mr~j- 

Yr»Wrcos  0  )     (7) 


»  Pe  +Xrr  -Yr  z 

- 


But  Xr-Wrsin  0  =  P-Mr  777  =Ka  is  the  resistance 


to  recoil  during 
the  accelerating  period,  and   2 

Xr-Wr  sin  0  =  -  Mr  -jpr  =  Kr  ia  the  resistance 

__  to  recoil  during 

the  retardation  period.   In  general  Ka  and  Kr  are 


137 


Fig. 3 


138 


different  in  value.    Hence,  let  K  =  Xr~Wrsin  0  for 

any  given  displacement  x  of  the  recoil. 

If  we  now  consider  the  reaction  of  the  recoiling 

parts  on  the  carriage  proper,  the  moments  about  the 
binge  A  of  this  reaction,  becomes, 

lp-  x  cos  0 

Xr(d-r)-Yr( +  d  tan  0-z)+Mr=ZMra 

cos  0 

Inserting  values  for  Yr  and  Mr  from  the  equations  of 
notion  of  the  recoiling  parts,  we  have: 

Xrd-Wrlr+Wrx  cos  0-Wrsin  0d+Pe  =  2Mra 

hence, 

(XrWrsin  0)d-Wr(lr-x  cos  0)+Pe=  2Mra   ) 

or  ( 

Kd-Wr(lr-X  cos  0)+Pe=  SMra  ) 

which  is  the  same  expression  as  obtained  before. 

We  may  regard  the  line  of  action  of  Xr  and  Yr 
to  pass  through  the  center  of  gravity  of  the  re- 
coiling mass,  together  with  a  couple  M=Pe,  the 
powder  pressure  couple  about  the  center  of  gravity 
of  the  recoiling  parts.   See  fig. (4) 

Taking  moments  about  A  we  have  directly 

1  -  x  cos  0 

Xrd-Yr( )  +  d  tan  0)+Pe  =  2Ura  (8) 

cos  0 

and  since  Yr=  Nr  cos  0.  we  have  as  before 

(Xr-Wrsin0)d+Pe-Wr(lr-x  cos#)=  2Mra        (9) 

where  JCr=B+R 

The  object  of  this  analysis  has  been  to  show  that 
so  far  as  the  external  effect  of  the  reaction  of  the 

recoiling  parts  on  the  carriage  mount  is  concerned 
the  exact  location  of  rod  pulls  or  the  lins  of  action 
of  the  guide  frictions,  is  entirely  immaterial,  though 
as  we  shall  see  immediately,  the  value  of  R,  the  sum 
of  the  guide  frictions,  does  depend  upon  the  line  of 
actions  of  these  pulls  together  with  the  friction  line 


139 


140 


of  action  of  the  guide  friction  itself  and  thus  in- 
directly the  external  effect  on  the  carriage  mount  is 
affected  slightly. 

Further  the  location  of  the  center  of  gravity  of 
the  recoiling  parts  may  considerably  change  the  guide 
frictions  during  the  accelerating  or  powder  pressure 
period. 

BRAKING  PULLS         The  total  resistance  to  recoil 
if  assumed  constant  throughout  the 
recoil  is  readily  evaluated  from  the 
following  relations: 
If  K  =  total  resistance  to  recoil 

(assumed  constant)  (Ibs) 

b  =  length  of  recoil  (ft) 

Vf=  velocity  of  free  recoil  at  end  of  powder  period 

(ft/sec) 
B  =  displacement  of  free  recoil  at  end  of  powder  period 

(ft) 
T=  time  of  powder  period  (sec) 

Then  from  the  energy  equation  for  the  movement  of 
the  recoiling  parts  after  the  powder  period,  we  have, 


(ft.lbs) 


Simplifying, 


K  =  «  mr  f  (Ibs) 

b-E+VfT 

With  a  variable  recoil  consistent  with  a  stability 
slope  "m",  and  assuming  a  constant  resistance  during 
the  powder  period,  we  have, 

If  K=  the  resistance  to  recoil  in  battery 

k=  the  resistance  to  recoil  out  of  battery 

instability  slope  =  °"r 

h    (h=height  of  axis  of 

bore  above 
ground) 


141 


2 

then  :^l{b-(E-  —-)]=  £  mr(Vf-~)*         (ft. Ibs) 


m 
and  k  =K-m[b-(E-  - — )]  (Ibs) 

Combining  and  simplifying,   we  have, 

K  =   ~L^~ ~* (lbs) 

2[b-E+VfT-  -  — (b-E)]  in   battery 

£  fflp 

at    ettikjj    si  :£!!•.•<>    r»i    V"^^*^09 

KT2 
k  =K-m(b-(E-  - — )]  out   of  battery 


II  )  , r  • 

B-the  total  braking  pull  (Ibs) 

R=the  total  guide  friction  (Ibs) 

Pk=tne  total  oil  pressure  on  the  hydraulic  pis- 
ton (Ibs) 

p' 

rj)=the  hydraulic  reaction  plus  the  joint  frictions 

(stuffing  box  +  piston)  (Ibs) 

Pa=the  total  elastic  reaction(due  to  compressed 

air  or  springs)  (Ibs) 

p' 

ra=the  total  elastic  reaction  plus  the  joint 

frictions  (stuffing  box  +  piston)    (Ibs) 
fti=the  normal  front  guide  reaction      (Ibs) 

^»=tbe  normal  rear  guide  reaction       (Ibs) 
u=coef f icisnt  of  guide  friction  (0.15  to  0.25) 

Then  K=B+R-Wr  sin  0   (Ibs)  Total  resistance  to  recoil 
where 

B=P'  +  P'        (Ibs)  Total  braking 
h    a 

R=  u(Qt+Q8)      (Ibs)  Total  guide  friction 

The  stuffing  box  friction  is  usually  assumed  at 
from  100  to  150  Ibs.  per  inch  of  diameter  of  rod,  and 
if  du  and  da  are  the  stuffing  box  diameters  of  the 

hydraulic  and  air  cylinders  respectively,  we  have 


142 


(Ibs) 


Pa=Pa+100da    (Ibs) 


GUIDE  OR  CLIP  REACTIONS     The  recoiling  mass  is 

constrained  to  translation 

V  <•  C 1 1 

parallel  to  the  axis  of  the 
jc.  3  (i  r 

"bore  by  the  recoiling  masses 

engaging  in  suitable  guides  in 

the  cradle  of  the  top  carriage.   In  general  the  re- 
coiling mass  may  recoil  in  a  sleeve,  a  part  of  the 
cradle,  or  along  guides  considerable  below  the  axis 
of  the  bore  and  the  center  of  gravity  of  the  recoiling 
parts . 

For  the  former  case,  considering  the  external  re- 
actions on  the  recoiling  mass,  fig. (5) 

aa-at=Wr  cos  0 

Q  xl  +  &8x2+u  (Qtyt-Q-2y2)-Fe-B  b=0    (moments   about 

the   center  of 
gravity   of   re- 
coiling  parts) 
where   eb=  d-db    ,    then  &txt+(at+llr   cos   0)+u  (Q^-Q^y.,- 


cos  0)-"Fe-Bb  =  0 
cos  J0(x2~uy2)-  Fe-Bb=0 

Hence          Fe+Beh-W_   cos   0(x  -uy    ) 

Qt= — * (10) 

Further 

Fe+Beh-W_   cos   0  x   +W_cos   0  uy   +Vlrcos   0   x   +W_cos   0  x   + 

U*  2*  2'  1*  2 

Wr  cos  0   uy  -Vf.cos  0  uy 

i  'II  "2 


143 

0(x  +uv  ) 
Hence  Q2=  -  -  -  —  (11) 


When  the  guide  reactions  are  below  the  axis  of  the  bore 
as  in  Fig.  (4)  y2  remains  the  same  in  the  above  formulae 
whereas  yt  reverses  in  sign.  Hence  for  case  (2) 

Fe+Beb-W_cos  0(x  -uy  ) 
Q=  -  —  -  -  -  -  2_  (12) 


and      Fe+Beh+HLcos  0(x  -ay  ) 

ft=  -  loti  -  *  -  L_  (13) 


The  total  guide  friction  becomes,   R=«(Q1+Q8) 
hence 


2(Fe+Beb )-Wrcos  0  x2+Wrcos   0   ny2+  ¥rcos 
xt   +     x2  +   u(yt   -  ya) 

2(Fe+Beh)+Wrcos   0f (x  -x  J  +  ii(y  +y    )] 


(14) 


x  +x  +.u(y  -y  ) 

1    2    w  1  '  S 

Now  if  M=x1-t-x,+  u(y1-y,!)  for  case  I 
or     M=x1+xs-u(yi+ya)  for  case  II 

and  if  N=(xt~X2)+  u(yt+ye)  for  case  I 
N=(xi-x2)+u(y2~y1)  for  case  II 

we  have  therefore,  in  general  that 

2(Te+Beb)+Wrcos  0  N 

R=  -  -  -  -  -  u  (15) 

M 

which  gives  the  total  guide  friction.   The  value  of 
the  coefficient  of  friction  u  ranges  from  0.15  to 
0.20. 

The  total  braking  evidently  becomes, 


2(Pe+Beb)+Wrcos   0  N 
B+   -    u-K+Wr   sin  0 

M 

or  B(M+2uh)=(K+Wr  sin  0)M-(2Pe+Wrcos  0  N)u 
hance    (K+Wrsin  0)M-(2Pe+Wr  cos  0N)u 

BS  . 

M+2Uh 

which  gives  the  total  recuperator  reaction  in  terms 
of  the  total  resistance  to  recoil. 

Denoting  as  before  by 

Pjj=  the  hydraulic  reaction  plus  the  joint  frictions 

(stuffing  box  and  piston) 
P0  ~  the  total  elastic  reaction  plus  the  joint 

3 

frictions. 
6^=  distance  from  center  of  gravity  of  recoil- 

ing parts  to  line  of  action  of  hydraulic 

brake  pull  Pn 
ea=  distance  from  center  of  gravity  of  recoil- 

ing parts  to  line  of  action  of  the  re- 

cuperator reaction  Pa 
The  front  and  back  clip  reactions  become, 


,  a      a  r  ,g  ( 


and 

cos 


(17) 


y^  reversing  in  sign  when  the  guide  reactions  are 
entirely  below  the  axis  of  the  bores.  Combining 
as  before  and  noting  that  F=u(Q  +Q  )  we  have 

t     2 

2Fe+22p'  ea+2ZPu  eh+Wr  cos  0  N 

R-  — h   h   r (18) 

M 
where  M  and  K  are  the  constants  referred  before. 

Now  K=  £Pa+2Ph+R-Wr  sin  0  (19) 


145 
and  combining  (18)  and  (19)  we  get 


KM=(2Pa+2Ph  -Wrsin  0)M+  n(2Fe+22Paea+22Pheh+Wrco5  0  N) 

Simplifying, 

KM=((2Pa-Wrsin0)M+ZP,!|  x  M+  u(2Fe+22Paea+Wrcos  0 

or  further  simplifying, 


a+Wrcos  0  N) 
Hence,  M(K_sp'+Win  0)-u(2Fe+2XP'ea+N   Wnos  0) 

I  at  0  a 


M+2U6J, 

which  gives  the  gross  hydraulic  pull  in  terms  of  the 
total  resistance  to  recoil,  the  gross  air  or  spring 
reaction  and  the  maximum  powder  force. 

APPROXIMATE  FORMULAE      Assuming  the  reaction  be- 
GUIDE  FRICTION         tween  the  recoiling  parts  to 

be  equivalent  to  a  normal 
force  passing  through  the 
center  of  gravity  N,  a  couple 

M,  and  the  braking  and  guide  friction  forces  B  and  R 
having  moment  arms  about  the  center  of  gravity  of  the 
recoiling  mass  equal  to  dv  and  r  respectively  where 
r  is  the  mean  distance  to  the  guids  frictions,  we 
have,  for  moments  about  the  center  of  gravity  of  the 
recoiling  parts, 
Be^+Rr=M  neglecting  the  powder  effect  which  is 

usually  very  small  and  N=Wr  cos  0  for  the 
total  reaction. 

Obviously  the  actual  normal  guide  reactions, 
becomes, 

0  x 


and       .  W_  cos  0  x. 

fl  s  -M+ i. 

*   1      1 


where  1  =  x  +x   also  R=u(N  +N  ) 

12  12 


146 

2M+lfrcos  0(x  -x  ) 
hence  R=u  ( • — ) 

Substituting  the  value  of  M,  we  obtain, 
u  Wrcos  0(xt-x2) 


(21) 


l-2ur 

Very  often  xl~xi  is  small  and  in  a  preliminary  design 
xt  nay  be  assumed  equal  to  xe 
Hence   2ufieb 

:  l-2Ur  (22) 

which  gives  an  approximate  value  of  the  guide 
friction,  useful  in  a  preliminary  design  -  u  may  be 
assumed  from  0.13  to  0.25. 

Very  often  as  in  symmetrical  barbette  mounts, 
the  value  of  Beb  may  be  small  due  to  a  small  value 
of  eb  and  a  certain  limitation  arises  as  to  the 
use  of  the  friction  formula  previously  derived. 

When  lf  cos  0  x    * 


1  1 

that  is,  Wrcos  0  xf  =  Be0+Rr  =  8eb  approx.  we  have 

continuous  contact  along  the  guides,  the  distributed 
guide  reaction  oalancing  the  weight  comoonent  normal 
bo  the  guides. 

For  such  a  condition  the  guide  friction, 
becomes, 

R=0.2  to  0.3Wrcos  0  (23) 


INCREASE  OF  GUIDE  FRICTION     If  we  assume  the  total 
DURING  POWDER  PRESSURE      oraking  B  to  be  constant 
PERIOD.  during  the  powder  pres- 

sure period,  the  guide  frict- 
ion R  is  augmented  by  the 

powder  pressure  couple  together  with  the  increased 
friction  couple. 

Let  B  =  the  constant  braking  force 
fa  the  varying  powder  force 


147 


Nt=  the  normal  reaction  of  the  stationary  part 

on  the  recoiling  mass. 
R  =  the  guide  friction  during  the  powder  pressure 

period. 
M  =  the  reacting  couple  of  the  stationary  part 

upon  the  recoiling  mass. 
RS  and  M2  are  the  corresponding  values  during  the 

retardation  period. 
Then,  during  the  powder  pressure  period,  we  have 


N-Wr  cos  0=0  (24) 


and  during  the  retardation  period,  we  have, 
-Wrsin  0=Mr 


d'x 


dt« 
N-Wr  cos  0=0  (25) 


Further,  let  AM=M1-M3  and  AR=Rt~R2  ,  then  subtracting 
(25)  from  (24)  we  have  AM=Fe+AR  r  (26) 

Now  during  the  accelerating  period  the  normal 
guide  reactions,  become, 

,   Mt    Wrcos  0  xf 


(27) 

U  '  -  *«-       KrCOS  0  Xi 


and  during  the  subsequent  retardation 

142   Wrcos  0  x? 

W*=  T       1  (28 

M2   Wr  cos  0  x  i 

Na=  T~   "T~ 


148 


Adding  the  two  equations  in  (27)  and  (28)  respect- 
ively and  subtracting  (28)  from  (2?)  and  multiplying 
by  the  coefficient  of  friction  n,  we  obtain  obvious 
expression: 

2AM 

*"•  T  U  (28) 

Substituting  (29)  in  (26),  vie  have 


AM=Fe+     ur  (29) 

Fe      Fel 
and  AJ4-  -  =  - 

1_  iHE   1  -2ur  (30) 

1 

and  substituting  in  (29),  ire  have  for  the  change  of 
friction  during  the  powder  period, 

(31) 


Thus  the  guide  friction  is  continuously  augmented 
always  proportional  to  the  total  powder  pressure, 
providing  the  braking  is  assumed  constant.   We  also 
note  an  additional  cause  of  first  class  importance 
for  the  reduction  of  "e",  that  is,  the  importance  of 
locating  the  center  of  gravity  of  the  recoiling  mass 
along  the  axis  of  the  bore. 

Another  cause  for  a  change  in  guide  friction  during 
the  powder  period  is  due  to  the  torque  reaction  of 
the  rifling,  Tr  though  the  total  guide  friction  remains 
the  same. 

The  normal  reaction  on  the  left  guide,  becomes, 


m  M_   wrcoa  0*a     Tr 
Ntl3  2,  "  2dg 


(32) 


149 


and  the  same  for  the  right  guide,  becomes, 

U      W-COS  0X.,     Tr 

N  s  * E 1  +  -£_  (33) 

*r   2X      2l       2dg 

Wrcos  0xt    Tr 
2l      '  2dg" 

where  dg  is  the  distance  between  guides. 

In  the  gun  recoiling  in  a  sleeve  this  torque 
must  be  balanced  by  the  reaction  of  the  key  way. 

Noting  that,  R-w(Ntl*Ntl  +  N  +Htr)»u(Nt  +  Nt) 

the  total  friction  remains  the  same. 

It  is  important  to  note  that  the  friction  on  the 
left  guide  over  that  on  the  right  due  to  this  rifling 
introduces  a  couple  in  the  plane  of  t"he  guides  which 
tends  to  cause  rotation  about  an  axis  normal  to  the 
plane  of  the  guides.  Therefore,  it  is  always  essential 
that  small  side  grooves  or  flanges  on  the  clips  be 
introduced.   The  additional  friction  on  the  flanges 
is  entirely  naglsgible,  but  the  normal  reaction  to 
the  flange  in  extreme  cases  may  be  considerable. 

During  the  recoil  the  guide  friction  is  seldom 
constant  since  the  distance  between  clip  reactions  pro- 
gressively decreases  in  the  recoil,  that  is  the  front 
clip  approaches  the  rear  part  of  the  guide  in  recoil. 
When  the  recoil  is  long  it  is  desirable  on  field  carriages 
to  have  an  additional  clip  near  the  muzzle  which  engages 
in  the  guide  sometime  later  in  the  recoil.   Due  to  this 
cause  the  guide  friction  continually  increases  until 
the  engagement  of  the  outer  clip  and  then  we  have  a 
sudden  drop  in  the  magnitude  of  the  friction. 

When  the  braking  pull  remains  constant  and  the 
powder  pressure  coupls  is  small  and  n o outer  clips  are 
introduced  during  the  recoil,  the  clip  reactions  should 
always  be  designed  for  the  condition  of  out  of  battery. 

To  recapitulate  in  the  limitations  in  design  so  far 
as  guide  is  concerned,  we  note, 


150 


(1)  The  bearing  pressures  and  consequent 
friction  of  the  guide  are  reduced  by  increas- 
ing the  distance  between  the  clip  reactions 
nearly  directly,  consequently  for  a  given 
guide  reaction  and  friction,  we  have  a 
minimum  distance  between  clip  contacts 

on  the  guides. 

(2)  The  guide  reactions  are  reduced  by 

bringing  the  resultant  of  the  rod  pulls 
through  the  center  of  gravity  of  the 
recoiling  parts. 

(3)  The  moment  effect  and  consequent 
guide  reactions  are  further  reduced  by 
bringing  the  resultant  guide  friction 
line  through  the  center  of  gravity  of 
the  recoiling  mass. 

(4)  It  is  highly  desirable  to  center  the 
center  of  gravity  of  the  recoiling  mass 
midway  between  the  guide  reactions. 
This  condition  is  usually  impossible  to 
attain  especially  out  of  battery,  but 
may  be  compensated  by  increasing  "1" 
the  distance  between  the  clips,  by  an 
additional  front  clip  near  the  muzzle. 

(5)  Proper  functioning  of  the  recoil 
may  be  srrtirely  destroyed  by  having  the 
center  of  gravity  of  the  recoiling  mass 
too  far  below  the  axis  of  the  bore,  thus 
introducing  a  powder  pressure  couple 
with  excessive  guide  friction  during  the 
powder  pressure  period.   This  powder 
pressure  couple  may  cause  a  "springing" 
of  the  guides  and  considerable  heating 
as  well.   The  center  of  gravity  of  the 
recoiling  mass  should  never  exceed  1.5" 
from  the  axis  of  the  bore  unless  a  friction 
disk  for  rotation  during  recoil  about 

the  trunnions  is  introduced. 


151 


COMPUTATION  OP  BRAKING      We  have  seen  from  the 
POLLS  previous  discussions  that 

the  guide  friction  is  not 
independent  of  the  braking 
pulls  due  to  the  hydraulic 

and  recuperator  reactions.   These  pulls  tend  to  cause 
rotation  and  thus  augment  the  guide  friction  over 
that  due  to  the  weight  component. 

The  total  resistance  to  recoil  is  given  by:- 
(1)     when  constant  during  recoil, 


(lbs) 


b-E+VfT 
where  Vf=  maximum  free  recoil  velocity   (ft/sec) 

T  =  total  powder  period  (sec) 

E=free  recoil  displacement  during  powder 

period  (ft) 

b=  length  of  recoil  (ft) 
•r=  recoiling  mass 

(2)     when  variable  consistent  with  the 
stability  slope  "m", 

mrVc+m(b-E)2 
K=  -  ;  -        (Ibs) 

2[b-E+VfT-  ^—  (b-E)] 
2  mr 

where 

K=B  +  Rg-Wrsin0=Pn+Pa+u(Q1-»-Q2)-Hfrsin0  (Ibs) 

and     8=Pn+Pa=   total  braking 

Rg=u(Ql+Q9)=   mean  guide   friction   assumed   constant 

2uBeh 
We   have   seen  -—  Ibs.approx. 


where  nf=0.15  to  0.2 

e)j=  distance  from  center  of  gravity  of  recoiling 
parts  to  line  of  action  of  B.   (in) 


162 


B3  total  hydraulic  and  recuperator  pull  (Ibs) 

I3  total  distance  between  clip  reactions  (in) 
r=  distance  from  center  of  gravity  of  recoiling 

parts  to  mean  friction  line  (in) 
then, 

2uBeb 

'   -  "  (Ibs) 


hence, 

(K+«_sin  0)(l-2ur) 
B«     T  -  - 
l+2u(eb-r) 

Further  since, 


(Ibs) 


8in 


we  have  on  simplifying 
~~J""  (K+Wrsin  j0)( 


(Ibs) 


l-2u(eb-r) 

Very  approximately, 

Rg=0.3  Wr  cos  0 
and 

B=K+Wr  sin  0-0.3  Wr  cos  0  (Ibs) 

Pn»K+Wrsin  0-Fa-0.3Wr  cos  0  (Ibs) 

INCREASE  OF  RESISTANCE  TO        During  the  powder 
RECOIL  DURING  POWDER  PERIOD   period,  the  powder  pres- 
sure couple  may  be  suf- 
ficient to  cause  a  large 
increase  in  the  guide 

friction,  whereas  the  braking  pulls  due  to  the  hy- 
draulic resistance  and  recuperator  reaction  are  not 


153 


affected.    Prom  the  previous  discussion  on  guide 
friction,  the  increment  in  guide  friction  equals, 

2Feu 

(lba) 


or  more  exactly 

2Feu 
2   M 
where 

M=xl+x2+u(yt-ys)     Guides  above  and  below 

axis  of  bore 

=xt+x2-u(yt+y2)     Guides  entirely  below 

axis  of  bore. 

Hence  the  total  resistance  to  recoil  becomes  during 
the  powder  period, 

K*=K  +  ARg  (Ibs) 

and  this  value  should  be  used  in  the  computation  of 
the  trunnion  and  elevating  gsar  reactions. 

Strictly  speaking,  the  value  of  K   is  slightly 
high,  since  the  augmented  friction  due  to  a  large 
powder  pressure  couple,  will  diminish  the  maximum 
velocity  of  restrained  recoil  and  thus  the  resistance 
to  recoil  for  a  given  displacement  b. 

A  more  exact  value  of  the  resistance  to  recoil 
can  be  estimated  as  follows: 
Thus, 

(K+ARtf)T*    i        (K+ARff)T 

K[b-(E — )]=  -  mr[Vf * — ] 

2mr        2          mr 

Simplifying,  we  have, 

*   r  f   *'  J    2m_ 

R.  ?—   (Ibs) 


154 


•bore  2uPae  2nFme 

£Rtf»  —————  -  -        (approx)(lbs) 
"»+x+u(-y)  l-2ur 


(lbs) 


Fro«  interior  ballistics,  we  have, 

'  w     aVu 

P«*«  1.12  - 


«     (b'+u0)8 

w  =   weight   of  projectile 
(Ibs) 
uo-    total   travel   up  bore 

(ft) 
v  =   muzzle   velocity (ft/sec) 


b'-u0[(|l-*-l)±/(l-ll-)a-l]  <"> 

~  (ft/sec) 

u 

where  P,=  total  maximum  powder  force        (Ibs) 

Unless  the  powder  pressure  couple  is  excessive, 
that  is  the  center  of  gravity  of  the  recoiling  parts 
is  considerably  below  the  axis  of  the  bore  the  above 
refinement  in  calculation  is  unnecessary.   When  e 
exceeds  1.5  to  2  inches  the  above  effect  becomes  of 
consideration. 

INTERNAL  STRSSS  IN  THE        It  is  very  important  to 
RECOILING  PARTS         observe  that  the  braking 

force  8  when  treating  of  the 
external  forces  on  the  re- 
coiling masses  as  in  the 

previous  discussions  refers  always  to  the  reaction  of 
the  oil  in  the  hydraulic  brake  and  the  spring  or  com- 


155 


pressed  air  reaction  of  the  recuperator.  During 
the  accelerating  period  the  reaction  on  the  gun  lug 

nay  differ  considerably  from  the  braking  force  B  due 
to  the  acceleration  of  the  piston  and  rods  where  these 
recoil  with  the  gun  or  to  the  acceleration  of  the  re- 
cuperator sleigh  or  slide  when  the  sleigh  recoils  and 
the  rods  are  fixed  to  the  carriage. 

If  now  we  consider  a  recoiling  mass   consisting 
of  a  gun  together  with  a  single  cylinder  recoiling 
with  the  gun,  figure  (4)  and  if  we  let, 

B3  the  total  braking  force  along  the  axis  of  the 

cylinder. 
B  =the  normal  reaction  of  cylinder  on  the  gun  lug. 

j  =the  tangential  or  shear  reaction  on  the  gun  lug 
M  -the  bending  moment  reaction  on  the  gun  lug. 

Neglecting  the  guide  friction,  let, 

QI  and  QS  be  the  normal  guide  reactions 

xt  and  x?  the  coordinates  along  the  axis  of  the 
bore  of  the  clip  reactions  with  origin 
at  the  center  of  gravity  of  the  gun. 

x  and  x   the  coordinates  parallel  to  the  axis 

of  the  bore  with  origin  at  the  center 
of  gravity  of  the  recoiling  parts. 
xc  and  yc=the  coordinates  of  the  center  of 

gravity  of  the  recoiling  cylind«r  with 
respect  to  the  center  of  gravity  of  the 
gun  as  origin. 

e^-  distance  from  the  center  line  of  the  recoil 
cylinder  to  a  line  through  the  center  of 
gravity  of  recoiling  parts  parallel  to  the 
axis  of  the  bore. 
.Mr  and  Wr=  mass  and  weight  of  the  recoiling  parts. 

Mc  and  tfc*  mass  and  weight  of  the  recoiling  cy- 
linder. 

j  =  the  distance  from  the  shear  reaction  on  the 
gun  lug  to  the  center  of  gravity  of  the  gun 


156 


itself. 

F»  the  maximum  powder  force  along  the  axis  of  the 
bore. 

If  the  mass  of  the  lug  is  negligible  as  compared 
with  the  mass  of  the  gun,  the  coordinates  of  the  center 
of  gravity  of  the  recoiling  mass  with  respect  to  that 
of  the  gun  becomes 

Vc           Vc 
xr»  -r  —  and  yr=»  e=  —  — 

"r  "r 

Further 


*t  -  -  -  and  y    y  -  -^j— 
"  " 


«.-  s  *  —  and  *.  3  ^ 

Now  considering  the  gun  with  its  lug  alone,  the 
reaction  of  the  recoil  cylinder  on  the  lug,  consists 
of  the  pull  B  a  bending  moment  Wc  cos  0  (xc  +  j  )  and 
a  shear  reaction  Wr  cos  0. 

Taking  moments  about  the  center  of  gravity  of  the 
gun,  we  have  fi'(b+e)  *  \»c  cos  A  j-Wc  cos  0(xc+j  )aQ8x^* 


B'eb*B'e-Wcco.  0  xc=Qt(xt+  -~^)*(ar 

wr  wr 

Simplifying,  we  have, 

B'eb+B'e  =  Ot(xt+xt)  +  Wr  cos  0  xa 

Considering  the  recoil  cylinder  alone  we  note 
that  during  the  accelerating  period, 

d'x 

B     -B+Wcsin0  »MC  - 
°  df 


157 


but  from  the  recoiling  mass,  we  have, 
d*x    F-B+Wr  sin  0 


dt«  Wr 

hence  B'  =  B-Vfc   sin  0  +  r-(F-B+Wr   sin  0)   and   substituting 

in  the  previous  equation,  we  have, 

V 
(B-Wcsin  0  +  --(F-B+Wrsin  0)(eb+e)=Qt 

"r 

"r5 
but  yc=   8t>+«   and  yr=   e  hence    (ejj+e)=  - — 


'r 

Wre 

"c 

Therefore,  substituting  in  the  above  equation,  we  have, 
8(eb+e)-Wre  sin  0+Fe-Be+Wrc  sin  0  =  Q1(xt+xt)+Wr  cos  £)  ; 
hence  Beh+Pe  =Q  (x  +x  )+W_  cos  0  x. 


Obviously  if  we  consider  the  recoiling  mass,  fig.  (4) 
we  have,  taking  moments  about  the  center  of  gravity 
X   Qx   tut  Q-Qs  W  cos  A 


Hence  Be^  +  Fe  -  Qt(xt+  xf)  +  Wr  cos  0  xa  the  same 

equation  as  obtained  above  as  of  course  we  should  ex- 
pect. The  above  discussion  shows  the  importance  of 
considering  either  the  mass  of  the  gun  with  its  proper 
external  reactions  or  the  mass  of  the  recoiling  parts 
with  its  proper  external  reactions  and  not  confusing 
the  mass,  of  the  gun  and  recoiling  parts,  and  the  co- 
ordinates of  their  center  of  gravities. 

The  maximum  stress  in  a  section  m  -  m,  see  fig.  (4) 
of  the  gun  lug  obviously  occurs  when  the  bending  moment 
due  to  the  weight  of  the  recoil  cylinder  is  a  minimum  and 
the  braking  force  B  a  maximum  that  is  at  maximum  elevation, 
In  the  above  discussion  the  normal  reaction  between  the 


158 


piston  surface  and  cylinder  was  assumed  zero.   This 
reaction  obviously  depends  upon  the  weight  and 
relative  deflections  of  the  rods  and  cylinders.   If 
these  weights  were  equal  and  at  the  same  distance  from 
the  point  of  support,  and  with  equal  elasticity, 
this  reaction  becomes  zero  and  we  have  the  bending 
moment  assumed;  but  since  the  rods  are  relatively  very 
elastic  as  compared  with  the  cylinder  in  general  the 
moment  Wc  cos  0  (xc  +  j)  if  anything  is  augmented. 

If  Imn  is  the  moment  of  inertia  of  the  section 
"«  -  m",  AJJ-U  its  area  of  cross  section,  and  y  is 

the  distance  to  the  edge  from  the  neutral  axis  of 
the  section  and  "g"  the  distance  from  6   to  the 
neutral  axis,  we  have  for  the  maximum  fibre  stress 


cos   0(xc   tJ)]y          fccos   0 

+  -    (34) 


c 

where  B  =B-Wcsin  0  +  —  (F-B+Wr  sin  0) 

wr 

Since  the  weight  components  are  small  as  com- 
pared with  the  powder  pressure  force  and  braking  for 
a  first  approximation,  we  have, 

wc 

[B+  —-(P-B)lg 
Wr 


_ 

a-n>=  (35) 

lm-m 

which  is  a  useful  formulae  for  practical  design. 

TIPPING  PARTS         The  tipping  parts  consist  of 

all  the  parts  that  move  in  elevation 
with  the  gun.   The  two  principje 
parts  of  the  tipping  parts  are  the 
recoiling  parts  and  cradle,  the 
one  moving  in  recoil  and  the  other  remaining 
stationary.   The  cradle  supports  by  its  guides  the 


159 


recoiling  parts  on  recoil,  it  takes  the  reaction  of 
the  braking  exerted  on  the  recoiling  mass  and  is  sup- 
ported by  trunnions  resting  in  bearings  in  the  top 
carriage  and  is  further  prevented  from  rotating  about 
these  trunnions  during  the  recoil  by  the  reaction 
between  the  elevating  pinions  of  the  top  carriage  and 
the  elevating  arc  of  the  cradle.   When  a  rocker  is  in- 
troduced between  the  elevating  pinion  and  cradle  for 
an  independent  line  of  sight  it  should,  properly  speak- 
ing, be  included  in  the  tipping  parts. 

It  is  of  fundamental  importance  to  always  balance 
the  center  of  gravity  of  the  tipping  parts  about  the 
trunnion  axis  since  with  massive  parts  the  elevating 
process  must  be  done  quickly  and  with  the  minimum  re- 
action on  the  elevating  pinion  of  the  top  carriage. 
Let  x  and  y  =  the  coordinates  parallel  and  nor- 
mal to  the  axis  of  the  bore. 

X  and  Y  =  the  x  and  y  components  of  the  trun- 
nion reactions. 

F  =  the  total  powder  pressure  force. 
E  =  the  reaction  between  the  pinion  and 

elevating  arc. 
j  3  the  radius  of  the  elevating  arc. 

6ea  the  angle  between  the  "y"  axis  and 
the  radius  to  the  elevating  pinion 

contact  with  the  elevating  arc. 

The  mutual  reaction  between  the  tipping  parts  and 
'top  carriage  may  be  divided  into  the  component  reactions 
X  and  Y  of  the  trunnions  and  the  elevating  arc  reaction 
E. 

By  D'Alerabert's  principle,  considering  the  inertia 
of  the  recoiling  mass  as  an  equilibriating  force,  we 
have  during  the  powder  pressure  period  assuming  the 
gun  practically  in  battery,  for  equilibrium  of  the 
tipping  parts,  that,  fig.  (5) 

,t 

F-Mr  — ~  -2X  +Wt  sin  0  +E  cos  Ge  *  0      (1) 

d  u 

for  motion  along  the  "x"  axis, 


160 


9  ~.s?.}  no  tsdiexe  fco 

-;*»oq 
for  motion  along  the  "y"  axis,  and         ^  »Mt,w 

d*x 
bnrp(®  +  s)-Mr  J^T   8-E  j  =  0 

For  moments  about  the  trunnions,  the  weight  of 
the  tipping  parts  having  no  moment  since  the 

center  of  gravity  is  at  the  trunnions  in  battery. 

1  i  •  •   • 

d*x 

But  F-Mr  T~T  =Ka  fcns  Votal  resistance  to  recoi|.,T 

*pj  ?.  1  -,..-,  .dur  ing  th8  accelerating  period. 
Hence  equation  (3)  reduces  to:  Pa  +  K.s-E  j  =  0  and  DO^ 
the  reaction  on  the  elevating  arc  in  battery  becomes, 


a  (  4  V  ' 

J        »fU    lc    eixs   5dJ    p4 
and   the    trunnion   reactions    in   battery,    becomes, 

(Fe+Kas) 
2X=Ka*Wtsin   0   *  cosee  )    • 

( 
fcn*   nointq   edJ    ne>ev?ift<j  isoiJoaci   edi    «   3  v 

(Fe+Kas)  ty  ^   (5) 

2Y*Wt  C03.0-  "~~~  '  sin  9  ( 


tb«   resultant   trunnion  reaction 


•y 

" 


S*  /  X*+Y*         making  ^tn  angle   tan"1  —   with   the 
"x"   attWfc  now   if,    fig.    (5) 


"rf^  =-'!l-0tal  weight  of  the  tipping  parts 
Wc  =  weight  of  the  cradle  :6f  the  tipping  parts 

Wr  =  weight  of  the  recoiling  parts 
3.T*  3d  ; 
.  It  *  fche  horizontal  distance  to  the  center  of 

• 

gravity   of   the    tipping  parts    (in  battsry) 

always    assumed   at   the   trunnions   from   the 

'  •  i.  f~  !  u  r  A  n  y  § 

hinge  point   of   the   top  carriage. 

•  To  account  for  contact  of  tooth  rack  with  the  pinion 
or  worm  of  the  elevating  gear,  the  reaction  B  makes  an 


gle  approx.  20°  with  the  tangent  to  pitch  lines, 
erefore  in  above  equations  (eg. 5)  *J  •  beoome«,"j  o 
•  and  "oos  Ce"  beooaes   »  oos   1C  +20;"  and  "«in  Q  " 


angle 

Therefore 

20 


beooaea  •«in(«t+20) 


161 


fcoJrqqii 

:    Js    si 


L-___i_i!L:_iW 

*+. 


iio^i  <?rfi  lo  v 


*— — "}'  -  V  Oi ^ 

s   s|*'i         v^r 


.  5 


162 


1^  *  the  horizontal  distance  from  hinge  point 

of  the  tipping  parts  when  the  recoiling  mass 

is  at  a  distance  "XH  out  of  battery. 
lc  =  the  constant  horizontal  distance  from  hinge 

point  of  the  center  of  gravity  of  the  cradle. 
Then  for  moments  about  the  hinge  point  in  battery, 

"t^t*  "r^r*  "c^c  and  for  a  displacement  "x"  of 
the  recoiling  mass  from  the  initial  position,  we  have, 
WtlJ  =  Wr(lr  -  x  cos  0)+«clc  Therefore,  the  moment 

of  the  tipping  parts  for  a  displacement  "x"  of  the  re- 
coil, becomes, 

Wt(lt  -  lj)  »Wr  x  cos  0)  (6) 

Hence,  for  any  position  out  of  battery  of  the  recoiling 
•ass,      i 

2X-MP  2-7  -Wt  sin  0  -E  cos  9e»  0       ) 

(   (7) 
2Y-Wtcos  0+E  sin  6e  =  0  ) 

for  motion  along  the  x  and  y  axis,  respectively,  and 

t(r  —  S+»r  x  cos  0  -E  j  =  0  (8) 

dt» 

for  moments  about  the  trunnion  axis.  But  the  total  re- 
sistance during  the  retardation  becomes, 

.» 
Kr-Mr   f  -B+R-Hr  sin  0  (9) 


Combining  and  reducing,  we  get  for  the  reaction  on  the 
elevating  arc,  for  a  recoil  "x" 


Kr  s+Wrx  cos 


and  the  trunnion  reactions  for  a  recoil  "x" 


(10) 


163 


(Rrs+Wrx  cos  0) 

2X»K  +Wt  sin  J6  +  -  :  -  cos  ee    )   -**' 

J  ( 

(Krs+Wr  x  cos  0)  )    (11) 

2Y=Kt  cos  0  -  -  ;  -  sin  ee      ( 

J  ) 

which  shows  the  trunnion  reaction  depends  only  on  the 
total  resistance  to  recoil  and  the  moment  effect  of 
the  recoiling  weight  out  of  battery. 

It  is  often  more  convenient  to  consider  the  reactions 
of  the  elevating  gear  and  trunnion  reactions  between 
the  tipping  parts  and  top  carriage  as  divided  into 
horizontal  and  vertical  components  rather  than  along 
axis  parallel  and  perpendicular  to  the  guides. 

The  elevating  gear  reaction  will  be  considered 
positive  when  the  line  of  action  may  be  resolved  into 
components  horizontally  to  the  rear  and  vertically 
upwards,  that  is  when  the  radius  joining  the  trunnion 
to  the  elevating  pinion  contact  with  elevating  rack 
is  measured  from  the  vertical  counter  clockwise. 
Galling  this  angle  ne,  we  have, 

9e=  0  +  ne   whereas  before  0=  angle  of  elevation. 

We  have  then  for  the  elevating  components,  measured 
horizontally  and  not  vertically. 

He  =  E  cos  ne 

Ve  =E  sin  ne 

and  measured  along  the  "x"  and  "y"  axis,  i.e.  along 
and  normal  to  the  axis  of  the  bore, 
X,»  =  E  cos 


YP=E  sin(0+n_) 


*  More  strictly  to  aooount  for  obliquity  of  tooth 
contact  of  elevating  me  o  h  a  n  i  3  n  ,  j  beoor.es  j  oos  20, 
003  0.  becomes  oos  (W  +20)  and  sin  O  becomes  s i n ( 0  * 2O ) . 

O  o  w 


164 


The  horizontal  and  vortical  components  of  the  trunnion 
reaction  become,  in  battery, 

(Fe+Ks) 
2H=Ka  cos  0+  —  —  -  cos  ne       ) 

J 

P  +K 
2V=Ka  sin  0+Wr  -  —  •  -  sin  ne 

J  \ 

.v.ieJJed    ito   Ji  ^  9dj 

and   out   of   battery, 

(Krs+Wrx  cos   0) 
2H»Kp  cos  0+  -  •  -  cos   ne 

fcnoi*   narii    ierij£-t    feSnenoqaioo   IfiaiJiev   bnadfiJocsi-jorf 
(Krs+Wrx   cos   0)  ) 


oi<"       ;  &•»  erf  ^s«  noiJcs  lo  snil  »riJ  ae« 

and  the  resultant  trunnion  reaction  becomes, 
no.  "ot  tuJ'fcsT  sHJ  aart*  ai  3&di 


S  =  j/p*  +  7* 

.*c 

INTERNAL  REACTIONS  OF       It  is  important  to  observe 
TIPPING  PARTS  ~         that  X,  Y  and  E  are  the  external 
ROCKER  INTRODUCED.      reactions  exerted  by  the  top 

carriage  on  the  tipping  parts 
which  include  a  rocker  if  used. 

The  total  reaction  on  the  trunnions  include  the  reaction 
of  the  top  carriage  X  and  Y  and  the  reaction  of  the 
rocker  Xr  and  Yr  .   Hence  the  resultant  reaction  on  th« 
trunnions,  become,  algebraically, 

l»oi 

X  =X+Xr     ) 

and  ((12)  =  the  shear  components  of  the 

V  =Y+Yr     ) 

trunnion  pins  on  the  cradle. 

The  reactions  on  the  rocker,  alone,  therefore,  become, 
the  reaction  of  the  top  carriage  pinion  E,  the  reaction 
of  the  trunnion  Xr  and  Yr  reversed  and  the  reaction 
of  the  cradle  M  reversed.   In  many  cases  an  elevating 
screw  is  used  between  the  cradle  and  rocker  and  when 
used,  M  reversed  is  the  reaction  of  the  elevating  screw, 


166 


•~— — ^______^  -:i     «• 

10    fi>.7^nT~-- 

;- 

fc{1  .*      £fl  *  '5  «  5  j_/ 

,13JJuOf     Sri-* 


/     / 


dd^     1C     8S»4*M!ibT003     "^"     bft*     "x" 

&sntiL'to  , *• 


.  J.-- 

aQn««si&   .!  .- 
*di   to 


'  ^    »*<!     *jV 

t  **  ,  -\  -  -,  -  -       -  \  rl    J 


2     0R     *     5l     STdHw 
•J    «Ml*    (.-- 


its  line  of  action  since  the  screw  is  hinge  jointed 
at  its  two  ends  necessarily  lies  along  the  axis  of 
the  screw.  The  screw  reacts  on  the  cradle  with  a 
force  +  M  and  is  under  a  compression  M  during  the 
recoil. 

Considering  now  the  equilibrium  of  the  rocker, 

we  have,  if 

• 

Wr=  the  weight  of  the  rocker 

k=  the  perpendicular  distance  from  the  trunnions 
to  the  line  of  action  of  M  which  is  the  line  through 

the  axis  of  the  elevating  screw  when  used,  or  the 
normal  to  the  contact  surface  of  rocker  and  cradle  when 
an  elevating  screw  is  not  used. 

B=  the  angle  between  the  line  of  action  of  M  and 

the  "y"  axis. 

xa  and  ym=  the  "x"  and  "y"  coordinates  of  the 
center  of  contact  of  rocker  with 
cradle,  or  hinge  joint  of  elevating 
screw  of  rocker  on  cradle. 
xr  and  yr  =  the  "x"  and  "y"  coordinate  of  the 

center  of  gravity  of  the  rocker  and 
is  measured  from  the  trunnions  towards 
the  breech  and  downward. 

For  equilibrium  along  the  "x"  axis  (see  fig.  (6) 


2Xr+E  cos  6  +  *r  sin  0~M  sin  8  =  0      (13) 
for  equilibrium  along  the  Y  axis 

2Yr+M  cos  B-E  sin  6C  **r  cos  ^  =  °      ^14^ 
and  for  moments  about  the  trunnion, 

Ej-Mk-W  (xr  cos  0-yr  sin)  =  0          (15) 

where  k  =  xra  cos  B+ym  sin  B.   It  is  often  convenient 

to  replace  Xr  cos  0-yr  sin  0  =hr  the  horizontal  distance 

from  the  trunnions  to  the  center  of  gravity  of  the 
rocker,  then  (15)  reduces  to 


167 


Ej-Mk-Wr  hr=  0  (15  ') 

hence   Ej_w'  h 

M=  (16) 

K 

and 

(Ej-w;  hr) 
2Xr=  sin  B-Wr  sin  0-E  cos  ee   ) 

k  ( 

>   (17) 

(Ej-Wr  hr)      < 
2Yr-E  sin  9  -Wr  cos  0  - cos  B  ) 

K 

which  shows  the  rocker  reactions  depend  only  on  the 
elevating  arc  pressure. 

If  now  we  consider  the  equilibrium  of  the  tipping 
parts  not  including  the  rocker,  we  have,  fig. (7) 

2J('=Kr+(Wt-Wr)  sin  0+M  sin  B  ) 

( 
2Y'»(Wt-Wr)  cos  0  -ti  cos  B  )  (18) 

and     Krs+Wrx  cos  0-V»'  hr 

M=  — — —  (19) 

k 

since  the  moment  of  (Wt~wr^  about  the  trunnions=  -Wr  hr 
but  now  from  equation  (12),  we  have 

2X=lfX'-2Xr     ) 
2Y=2Y'-2Yr     ( 

hence  substituting  the  values  obtained  in  (17)  and  (18), 
we  have, 


*       To  account  for  tooth  contact,  more  strictly,  re- 
olaoe  j  to  j  cos  0.;  oos  0.  to  oos.(0  +20)  and  sin.  0   to 


168 


w-t 


bnsqsb  eueJ:49£Ai 


won 


.  -.  o  ft.  r  i  fit .  "I 

.«  i  *     tn  &      (  Ot  » 


169 


(      2X=Kr+(Wt-tfr)    sin  0+14  sin:  B-<H  s&n  B-W     sin  £J-E  cos  6e) 
(      2Y=(Wt,-wr)   c^3  0"14  C50S  ?~(?  r*ae^-Wri  cos  0^»CoQS   B) 
Simplifying   these   values   reduce'  tV  the-  fofrnrer  Values, 


2X=Kr+Wt  sin  0+E  co«  ee 

.(v.TSJtfid  ni  nu§)  g^rseq. 

2Y=Wt   cos  0  -B   sin'0e' 
and   further 

sieriw   n.'4o   artj    \o    eujbfii    J  •.  1    *    ,1 

i  i 

Krs    +Wr   x   cos   0-IL.  hr  .  EjHlL   hr 


-  —  -  -  -     -  —  •  _ 

k  k 

.<asc   "io   asjibAf   n*eM  B  H 
and  .therefore   as.  befor*  ,0  floU36il9b     e   ^ 

.noi4*vele  cBorclxfi*   Je    gaii  ::Krs+Wr   x  cos  0 

B=  -  .  *> 
-*      -  iiu^nof;    gnt-rqe 

thus  checking  the  formulas  derived  for  the  rocker 

reactions. 

oa  be;..          boil\£o  s/iiiqe  8.Hj  ni  SfiiTqs  tsriT 
TRUNNIONS  LOCATED  AT       In  guns  shooting  at  -hijjjh  J 
THE  REAKr  BALANCING^  r.  ,elevAti<m  s,uc.A  ;^s;  aqti- 
GEAR  OR  EQUILIBRATOR..  .T.,7cair;cra=ft  ^ttn^,:  mortars,  and 

even  howitzers,  it  is  often 
necessary  to  Ipcate  the  trun- 

nions in  the  rear  or  near  the  breach  of  the  gun  in  order 
to  prevent  the  breech  of  the  gun  from  striking  the 
ground  during  the  recoil  when  the  gun  is  elevated.  Ob- 
viously it  is  impracticable  to  balance  the  tipping 
parts  about  the  trunnions  without  some  sort  of  a 
balancing  gear  commonly  known  however,  as  an  equilibrator 

An  equilibrator  should  balance  the  tipping  parts, 
since  the  center  of  gravity  of  the  tipping  parts  are  now 
displaced  forward  of  the  trunnions,  at  all  angles  pf 
elevation  when  the  gun  is  in  battery.   It  consists 
sometimes  of  a  cam  arc,  a  chain  passing  over  the  con- 
tour of  the  cam  arc  and  connected  with  a  spring 
cylinder  oscillating  about  the  trunnions  fixed  to  the 

top  carriage  to  take  care  of  the  small  deflection  due 


170 


to  the  change  of  radius  of  the  cam. 

PROCEDURE  TN  DS3TGN  OT  B B U  I  L I B R AT OR : 

Let  Wt  =  weight  of  the  tipping  parts. 

ht  a  horizontal  distance  from  the  trunnions 
to  the  center  of  gravity  of  the  tipping 
parts  (gun  in  "battery). 
ro  =  equivalent  radius  of  cam  at  horizontal 

elevation. 
rn  =  final  equivalent  radius  of  the  cam  where 

the  cam  arc  has  turned  through  the  maximum 
angle  of  elevation  =  0. 
B  =  Mean  radius  of  cam. 

dn  =  deflection  of  spring  at  zero  elevation. 
d0  =  deflection  of  spring  at  maximum  elevation. 
C  =  spring  constant. 

0  =  the  angle  of  elevation  expresses  in 
rad  ians . 

The  spring  in  the  spring  cylinder  is  arranged  so 
that  it  is  in  general  under  compression.   As  the  gun 
elevates,  the  compression  of  the  spring  is  decreased 
in  virtue  of  the  motion  of  the  cam.   The  total  motion 
of  the  spring  becomes, 

R0  '  dh~do  aPPro*'  during  the  elevation  0  where 

ro*rn 

"  for  a  first  approximation  and 

m 

c  dhro=  *tht  for  equilibrium  about 

the  trunnions  for 
C  dorn=  *tht  cos  0-         all  elevations. 

If  now  we  assume 

dh=(-  to  -)d  solid 

11     3        4 

and  i    t 

do  =(—  to  -g-)d  solid 

Then  for  a  preliminary  design,  we  have, 


171 


C  =  -i 5 

(-  to  -  )d  Solid  r0 


Wtht  cos  0 
C  = 

*         * 

(-  to  -)  d  solid  rn 

r  +  r 
and  (-  to  -)d  solid  -(-  to  j)d  solid  =  ( — - — )0 

0 

The  unknowns  in  the  above  equatibns  are  C,  d  solid  ro 
and  rn;  hence  if  we  assume  any  one  of  these  values 
the  remaining  values  are  determined. 

The  equivalent  radius  of  the  cam  may  be  obtained 
by  a  "point  by  point"  method  as  follows'- 
The  initial  radius,  becomes,     g  h 

"t   L 

r°*5*r 

Now  move  the  cam  an  increment  angle  A0  where  nA0=0  the 
total  angle  of  elevation,  and  we  have, 


cos  # 

and  Ad  =  rnA0 


C(dh-r0A0) 

then 

W^ht  cos  0 


**  C[d-(r0+rM07  >V<po+' 

Htht   cos  0 
rn~  C[dh-(r0+rt rn_,)A0]  dh~do=    ro+rt~"rn-i) 

A0 

Strictly  speaking  the  angle  A0  in  the  above  pro- 
cedure should  be  augmented  by  the  angle  r"n~  r° —  radians- 

Where  D  =  the  perpendicular  distance  from  the 
trunnions  to  the  extremity  of  the  equivalent  cam  radius. 

From  the  equivalent  radius  thus  obtained  the  cam 
contour  may  be  drawn  by  drawing  in  a  curve  always 
tangent  to  the 


172 


perpendiculars  to  these  radius,  drawn  from  their 

f. 
extremities. 

With  a  balancing  gear  or  equilibrator,  the 
trunnion  reactions  are  modified  and  now  become  ___  -  r 
it 

T  =  the  tension  in  the  chain. 

a  =  the  angle  T  makes  with  axis  "X",  (takea  *)  j 
along  the  axis  of  the  bore) 

Ks*WrXcos  0+Fe 

-a—  iui)deos  0e*T  cos  •  *>«3)»e-c 
onj-  \n&  ;,,^  bns 

> 


)sin6e   *T   sin   a        iT( 

J  I   v,d   Jni:oq")B 

and  soihsT    IsiJ/ni:    ftri'JY 

) 


j 
-f"'  ei^ns-Jnsaaeioni  ne 


i!»8 

Usually  d  =  0.  and  thus  the  Y  component  of  the 
trunnion  reaction  is  unaffected. 

DIRECT  ACTING  BALANCING       Another  form  of 

GEAR  balancing  gear,  for      nedj 

balancing  the  tipping 
'3+  'O~  i,Parts  at.  all  angles  of  <, 

'  elevation  about  the 

trunnions  which  are  located  to  the  rear,  consists 
of  a  spring  or  pneumatic  oscillating  cylinder  and 
its  rod  directly  connected  between  the  tipping  parts 
and  the  top  carriage. 

In  the  position  of  the  tipping  parts  at  zero 
elevation,  gives  maximum  moment  and  therefore  re- 
quires the  •axinua  balancing  reaction. 

$rtj  moil  ocf.flJaib  IB  i  •  jdW 

~  To  .ccount  for  tooth  contact  of  .levatl.g 
••ohaniSB,  replace  j  to  j  COS  20,  oos  S  to  oos 
(Ot«20)  »rd  .In  d,  to  .in(«e+20). 

•  n  - 

sri.t  OJ 


173 


M          ..-•?      .noiievele  ed.4  &i»jt«or>  «   ae   eseeio 

Wt   =   weiSnt   °f   th8    tipping  parts 

nt   =   horizontal  distance    from  the    trunnions    tcbijr;nu} 
the  center  of  gravity  of  —the  tipping  parts   -11^0 
%nii  el  j(g  un-  i^  b  *Vt  &ry.<\»  r,  o  x  n  n  01  4   eri  J    §fil,»f.! 
.u^jfc  *od  yt  -  coordinates   alon^   and   normal   to  bore     ; 
-oneisd   orjfiiiiuffroift-  trunjiion   to  center  of  gravity 

of   tipping   parts    (gun   in  battery:)i.2.rq   5jni 

0   -  angle   of  elevation.,  Juov.*!   VJ«nif«il9Tq  s   io1 
0m  =  maximum  elevation>..   erfj    Ic   sixs   oornnot^   arii    gni 
r  =   radius   from   the   trunnion   to  the  crank  pin 
bft!j,llE:f*hich  connects   the   tipping   parts   to   the  pi*-; 

vfciJfcon  rod  of   the  oscillating   cylinder,  (in.)  iBi^cni 
R   =  reaction   exerted  by   the  balancing  gear   along 
the  piston  rod   of  the  oscillating   cylinder;. 

s)  3   iscfnos  iiori   TO   Isi^iai    siU    : 

dt   =   moment   arm  of  R  about   trunnion.  (in) 

d      =  deflection   of  spring   at    horizontal   elevation. 


do  =  deflection  of  spring  at  maximum  elevation. 

(in)     ,ooi**veJs  uwsilxsa  icl  seasooed 
C  =  spring  constant. 

RJ  =  initial,  balancing  reaction  (0°  elev.) 
Rt  =  final  balancing  reaction  (0*  elev.)(lbs) 
S  =  stroke  of  piston  in  oscillating  cylinder.  (in) 
pt  -  final  ,air  piressure  j.^  pneumatic  balancing 

cylinder  (Ibs/sq.in.) 
,|»£  =  initial  air  pressure  in  pneumatic  balancing 

cylinder  (lbs/sq.in.1 

^,f,  effective   area  of  balaacing   pistoa  ..(sjj.>cinj._)  j 
V0  =   initial    air   volume    (cu.in.) 

*\L    ,'J3bniI^o   Jniiq*   6   ri;  :?•• 
At   any  angle   of  elevation  0,    we   must   have, 
R  dt   »    Et(xt    cos    J0-y-t    sin  0)  io? 

In  general   the  center  of  gravity  of   the   tipping 

parts    lies   approximately   along    the   axis   of   the   trunnions, 
and   therefore,    R  d      =  W     x     cos   0. 


174 


If  dt  remains  constant,  the  reaction  R  should  de- 
crease as  a  cosinef unction  in  the  elevation.   Since  it 
is  usually  impossible  to  decrease  R  according  to  a  cosine 
function,  we  may  so  locate  the  trunnion  of  the  oscillating 
cylinder  so  that  the  product  R  dt  =  Wtxt  cos  j0.   By 
properly  locating  the  trunnion  axis  of  the  oscillating 
cylinder  a  very  close  balancing  is  possible  throughout 
the  elevation  either  with  a  spring  or  pneumatic  balanc- 
ing piston. 

For  a  preliminary  layout,  we  may  start  by  locat- 
ing the  trunnion  axis  of  the  oscillating  cylinder 
somewhere  depending  upon  clearance  considerations, 
along  a  line  parallel  to  the  chord  joining  the  assumed 
initial  and  final  positions  of  the  crank  and  midway 
betwean  the  chord  and  middle  of  arc.   The  crank  turns 

an  angle  equal  to  the  total  elevation  & 

m* 

Then   the   initial   or   horizontal   balancing   reaction, 
becomes, 

r         0m 

R^  -  (1  +  cos  — r)=Wt  xt  and  the  final  balancing 
2  <c 

reaction,  becomes  for  maximum  elevation, 

r       *m 
Rf-(l+  coa  T~-)3Wf  (*t  cos  flfl+yt  sin  0m) 


3  Wt  xt  cos  tm   (approx.) 

Rf 
We  have, therefore, cos  0m   very  roughly,  and  the 

Ri  *• 

total  stroke  of  the  piston,  becomes,  S=2r  sin  -r—  (in) 

Now  with  a  spring  cylinder,  if, 

dh  s  3  to  '4   solid  height  of  spring   (in) 

do  =  -  to  -  solid  height  of  spring    (in) 
then 

dh-d0=  S    (in) 


175 


and  at  0°  elev.  R^  =  c  dh  at  max.  elev.  Rf=  c  do. 

Hence  the  required  spring,  may  be  approximated  by 
the  solution  of  the  following  equations, 

= =  cos  0m          ) 


dh-do=' 

21 

Cdh=- 


r(l+cos — )  ( 

• 

With  a  pneumatic  cylinder,  we  have,  since  the  expansion, 
may  be  assumed  isothermal, 

pf    vo 
——  =  — —  =  cos  J0jj,      (aoorox.)  from  which  we 

i|   vo+ 

may  determine  the  initial  volume  Vo   hence 


pi         cos  0^ 
V0  =  AS =  AS (approx.)   (cu.in.) 

*f       1-  cos  0m 


Now 

pf 

RiS  TT 

Ri  =  p^A  and  V0=  — -—  [ ]    (cu.in) 

Pi      PP 
1  --1 

Pi 


We  see,  therefore,  to  decrease  the  tulk  of  the 
cylinder  it  is  important  to  maintain  as  high  an 
initial  air  pressure  as  possible.   It  is  reasonable 
to  assume  the  same  initial  pressure  as  used  in  the 
recuperator  brake. 

Therefore,  in  the  first  approximation  of  the  de- 


176 


sign  layout  of  a  pneumatic  balancing  gear,  we  may 
start  with,- 


Pf 

(cu.in) 


V0   ' 

2WtxtS 

' 

PT 

0     v 

r(l+cos   -    )p; 
2        ] 

1  - 

Pf 

0   Pf 

where  S  *  2  r  sin  -  ,  —  =  cos  0m      (approx.) 
2   Pi 


REACTIONS  ON  TIPPING  PARTS     With  the  trunnions  to 
WITH  BALANCING  GEAR.        the  rear  of  the  center 

of  gravity  of  the  tipping 
parts  and  a  balancing  gear 
introduced,  we  have  a 

cantilever  form  of  top  carriage,  and  the  reactions  on 
the  tipping  parts  are  usually  approxircataly  in  the 
position  shown  in  fig.  (8). 
Let 
x  and  y  -  coordinates  along  and  normal  to  axis  of 

bore. 

R  »   reaction  of  balancing  gear    (Ibs) 
9r=   angle  between  R  and   y  axis. 
dts   moment   arm  of  R  about   the   trunnions   at   any 

elevation  0. 

E  =  elevating  arc  reaction   (Ibs) 
9e=  angle  between  the  y  axis  and  the  radius  to 
the  elevating  pinion  contact  with  the 
elevating  arc. 

J=  radius  of  elevating  arc  from  trunnions  (in) 
X  and  Y  =  components  along  x  and  y  of  trunnion 

reaction.  (Ibs) 
Wt  »  weight  of  tipping  parts, 
x^  and  yt  *  coordinate  of  center  of  gravity  of 

tipping  parts  from  trunnion  (in) 
Wr  =  weight  of  recoiling  parts  (Ibs) 
He  have,  then,  two  positions  to  consider,  the 
in  and  out  of  battery  positions  respectively.   In 


177 


JPEAC7/CW5  ON  T/PP/MG  PA/?TS  W/77J 
DIRECT  ACr/N6  BALAMC/MG 


TPU/W//OM  AX/S 


rig.  a 


178 


the  battery  position  we  have  the  effect  of  the  powder 
pressure  couple,  while  in  the  out  of  battery  position, 

we  have  the  moment  effect  of  the  recoiling  weight. 
Considering  the  Recoiling  parts  in  battery: 

We  have  for  the  kinetic  equilibrium  of  the  tipping 
parts,  for  motion  along  the  x  axis, 

d% 
Pb-  mr  —  —  -2X+Ht  sin  0  +E  cos  ee  +  R  sin  9r  =  0 

Q  X 

for  motion  along  the  y  axis, 

2Y-Wt  cos  0  -E  sin  6e  +  R  cos   6r  =  0 

and  for  moments  about  the  trunnion, 

,  2 
Pb(e+s)-mr—  7-S+R  dt-Wt(xtcos  0  -yt  sin  0)-Ej=  0 

d  t» 

Since,    however,  2 

pb~rar  =   K      and   R  d      = 


-y+  sin  0)  requirecl  condition  of  the  balancing  gear, 
the  above  equations,  reduce  to,- 

Ks+Pbe 
E=  -   (Ibs)  for  the  elevating  arc  re-   ) 

action  in  battery          ( 
2X=K+Wtsin0+E  cos  ee+R  sin  9r      (Ibs)  ) 


2Y=lft   cos   0+B   sin   9e  ~R  cos   er  (Ibs)  ) 

( 

Wtxtcos  0   2Wtxtcos  0  ) 

where  R  =  -  -  -  =  -  -  —    approx.(lbs) 

dt  0m  ( 

r(l+  cos  -  ) 

It  is  to  be  noted  that  2X,  2Y,  E  and  R  are  to  be 
regarded  as  the  reactions  exerted  by  the  top  carriage 
on  the  tipping  carts,  a  rocker  if  used  being  included 


*       To  account   for  tooth  contact  of  elevating 
••••••!••,   substitute  j  cos  20  for  j ,  B  co»(C.-20) 

for  I  cos  O.  and  E  sin(0  -20)   for  I  iin  0.. 

09  o 


179 


as  a  part  of  the  tipping  parts.   The  resultant  bearing 
reaction  between  the  top  carriage  bearing  trunnions, 
becomes, 

S  =  /  XZ  +  YZ       (Ibs) 
Considering  the  recoil  parts  out  of  battery 

if 

Wt=  total  weight  of  the  tipping  parts 
Wc=  weight  of  the  cradle  of  the  tipping  parts. 
Wr=  weight  of  the  recoiling  parts. 
xro  and  yro  =  battery  coordination  of  the  re- 
coiling parts  with  respect  to 
the  trunnions. 
xc  and  yc  =  coordinates  of  the  cradle  with 

respect  to  the  trunnions. 

Xfo  and  yfo  =  coordinates  of  the  tipping  parts 
in  battery  with  respect  to  the 
trunnions . 

For  a  displacement  "x"  of  the  recoiling  parts 
froB  the  initial  battery  position,  we  have,  for  moments 
about  the  trunnion, 

_To  .SEC- I  erii  oJ  v- rtiostu 

d  x 

+WC   sin   0  yc-E  j    =   0 

U  Utf     v     —     W     v         +W     v 

POW     n-fXj.1"     ffT%A_»-.~n.>A-, 


hence  the  moment  equation  about  the  trunnion  reduces  to 
,» 


mr  -  .t-txtcos   +tytsn   +rx  cos  0-Ej=  0 

QTf 

but  due  to  the  balancing  gear,  we  have, 


"txt  c°3  ~yi  s^n 

and  further  K=mr—  -  —  —  that  is,  the  inertia  resistance, 
d  t 

equals  the  total  resistance  to 


180 

recoil. 

Hence   K  s  +  Wr  x  cos  0 


For  motion  along  the  x  axis,  we  have,      ( 
2X=K+R  sin  er+E  cos  ee+Wt  sin  0  ( 

) 

and  for  motion  along  the  y  axis,  ( 

2Y=«ft  cos  0+E  sin  e+g  sin9s  -R  cos  er      ) 

I 

where   as  before  ; 

2Wt   xt   cos  0 
R=  roughly.  ) 

0o>  ( 

r(    1+cos     -— ) 

f  ) 

From  the  above  analysis,  we  see,  therefore, 
that  the  elevating  arc  reaction  remains  the  same 
with  or  without  a  balancing  gear,  while  the 
trunnion  reactions  may  be  increased  or  decreased 
according  to  the  location  of  the  line  of  action  of 
the  balancing  gear. 

INTERNAL  REACTIONS  OF       The  rocker  reactions  depend 
TIPPING  PARTS  WITH       solely  on  the  elevating  gear 
BALANCING  GEAR  -         reaction,  and  with  a  balanc- 
ROCKER  INTRODUCED.       ing  gear,  the  elevating  gear 

reaction  is  independent  of  the 

eccentricity  of  the  center  of  gravity  of  the  tipping 
parts  from  the  trunnions.   Therefore,  the  rocker  re- 
action on  the  trunnion  is  entirely  independent  of 
the  reaction  exerted  by  the  balancing  gear  or  counter- 
poise.  In  brief,  the  rocker  reactions  remain  the 


To  account  for  tooth  contact  of  elevating 
•echanisB,   substitute  j  cos  2O  for  ,i  E  oos(O.-20) 
for  B  oos  68  and  B  Bin  (O0  -  20)  for  B  sin  O°. 


181 


same  with  or  without  a  counterpoise  or  balancing 

gear.   The  reactions  exerted  by  the  top  carriage 
on  the  trunnion  do  however  depend  on  the  mag- 
nitude and  direction  of  the  balancing  gear. 
Therefore,  the  shear  and  bending  at  the  section 
of  the  trunnion  adjoining  the  cradle  must  also 
depend  on  the  balancing  gear  or  counterpoise  re- 
action. . 

An  analytical  proof  of  the  reactions  is  given 
as  follows:- 

Let 

X  and  Y  =  trunnion  components  of  the  reaction 
of  top  carriage. 

Xr  and  Yr  =  trunnion  components  of  the  re- 
action of  the  rocker. 

X   and  Y  =  shear  components  of  the  trunnion 
pins  on  the  cradle. 

Wr  =  weight  of  rocker. 

E  =  elevating  gear  reaction  on  rooker 

M  =  cradle  reaction  on  rocker. 

j  =  radius  to  elevating  gear  arc. 

k  =  the  perpendicular  distance  from  the  trunnions 
to  the  line  of  action  of  M  which  is  the  line 
through  the  axis  of  the  elevating  screw,  when 
used,  or  the  normal  to  the  contact  surface  of 
rocker  and  cradle  when  an  elevating  screw  is 
not  used. 

B  =  the  angle  between  the  line  of  action  of  M 
and  the  "y"  axis. 

xffl  and  ym  =  the  "x"  and  "y"  coordinates  of  the 

cradle  hinge  joint  of  rocker  elevat- 
ing screw  or  the  center  of  contact 
of  rocker  on  cradle. 
xr  and  yr  =  the  "x"  and  "y"  coordinate  of  the 

center  of  gravity  of  the  rocker  and 
is  measured  from  the  trunnions  towards 
the  breech  and  downward. 


182 


Evidently  for  the  shear  at  the  cradle  section 
of  the  trunnion, 

x'=  x  +  xr   ) 

( 
Y'  =  Y  +  Yr    ) 

For  the  angular  equilibrium  of  the  rocker, 
Ej-Hk-lfr(xr  cos  0  -  yr  sin  0)  =  0 

if  we  let  xr  cos  0-yr  sin  0  =  hr,  then 

Bj-Wrhr 
Ha  1 —  ,  where  k  =  xffl  cos  B  +  ym  sin  B,  that  is, 

the  cradle  rocker  reaction  depends  solely  on  the 
elevating  gear  reaction. 

For  the  translatory  equilibrium  of  the  rocker, 
2Xr=  M  sin  B-E  cos  ee  -Wr  sin  0 

2Yr=E  sin  (Je  ~Wr  cos  0  -M.  cos  B 

which  shows  the  rocker  reaction  at  the  trunnion  de- 
pends only  on  the  elevating  arc  pressure  and  there- 
fore is  independent  of  the  counterpoise  reaction. 

Considering  the  equilibrium  of  the  tipping  parts 
not  including  the  rocker,  we  have, 

2x'  =  Kr-i-(Wfc-1fr)  sin  0  +M  sin  B*R  sin  6r 

2Y*=(Wt-Wr)  cos  0-M  cos  B-R  cos  er 
Further  if  measured  from  the  trunnion  axis, 

lt=xtcos  0  -ytsin  0=  the  horizontal  distance  to  center 

of  gravity  of  tipping  parts(re- 
coiling  parts  in  battery) 
lt=  the  horizontal  distance  to  center  of  gravity  of 

tipping  par.ts  (recoiling  parts  out  of  battery) 
lr=  the  horizontal  distance  to  center  of  gravity  of 
recoiling  Parts  in  battery. 


183 


lc=  the  horizontal  distance  to  center  of  gravity  of 

cradle  . 
-hr=  the  horizontal  distance  to  center  of  gravity  of 

rocker  measured  in  a  negative  direction  from  the 

i's. 
d^  =  moment  arm  of  the  counterpoise  reaction  R  about 

the  trunnions. 

Then  since  the  moment  of  the  tiooing  parts  minus 
rocker  about  the  trunnions  is  equal  to  the  moment  of 
the  weight  of  the  tipping  parts  minus  the  moment  of 
the  weight  of  the  rocker,  we  have, 
W~h=?1~  x  cos 


Now     Ifrlr+Wclc=   «Ttlt+Wrhr 

hence  Wtl^+Wrhr=Wtlt  +Wrhr~Wr  x  cos  0 

=Wt(xt  cos  0-yt  sin  £J)  +  Wrhr~Wr  x  cos  0 
therefore, 
Krs+Rdt-Wt(xtcos  0-yt  sin  0)-l?rhr+Wrx  cos  -Wk=  0 

but  for  equilibrium  of  the  tipping  parts  in  battery 
Rdt=Wt(xt  cos  0-yt  sin  0) 


hence 


Krs+Wrx  cos  0-1 
M  =  • 


Since,  however,   2X=2X'-2Xr   ) 


2Y=2Y-2Yr 


We  have  in  substituting  the  previous  values  for 
2X*  and  ZXp^j, 

2X=Kr+Wtsin  0  +R  sin  9r+E  cos  6e 
2Y=lft  cos  0-E  sin  9e~R  cos  6r 


184 


Krs  +Wrx  cos  0 
and  E  =  — 


J 

In  the  preceeding  analysis  it  is  important  to  note, 
that  the  center  of  gravity  of  the  rocker  is  assumed 
to  the  rear  of  the  trunnions,  and  the  elevating  gear 
reaction  is  considered  positive  when  the  radius  to  the 
pinion   contact  of  the  elevating  rack  is  measured 
counter-clockwise  with  respect  to  the  "y"  axis 
through  the  trunnions.   Evidently  when  QQ  is  negative 
(i.e.  clockwise  from  "y"  axis,  E  cos  6e  remains  the 
same  but  E  sin  6e  becomes  negative  in  the  above 
equations. 

•ol' 

EFFECT  OF  RIFLING  Cue  to  the  rifling,  the 
TORQUE  ON  TRUNNION  torque  exerted  on  the  gun  by 
REACTION  the  shell  aust  be  balanced  in 

considering  the  equilibrium 
of  the  tipping  parts  by  an 

equal  and  opposite  moment  exerted  by  the  top  car- 
riage on  the  trunnions(assuming  due  to  the  ouch 
greater  flexibility  of  the  elevating  arc  and  pinion 
that  the  elevating  arc  reaction  is  entirely  unaf- 
fected). 

If  the  rifling  is  right  handed,  then  in  the  di- 
rection of  the  muzzle,  the  Y  component  of  the  left 
trunnion  is  increased  and  the  Y  comoonent  of  the 
right  trunnion  is  decreased  by  the  amount  equal  to 
the  torque  of  rifling  divided  by  the  distance  between 
the  trunnion  bearings  on  the  top  carriage.   Usually 
this  affect  is  quite  negligible  as  compared  with  the 
ther  forces  exerted. 

STRENGTH  OF  THfe  TRUNNIONS.     The  critical  section 

of  the  trunnions  is  usually 
where  the  trunnion  joins 
the  cradle. 

Let  "«n"  represent  this  section  on  the  trunnion, 

see  fig. (9). 


185 


xr.-s    no 


n 


186 


a  =  the  distance  from  "mn"  to  the  center  of 

the  top  carriage  oearing. 
b  =  the  distance  from  "mn"  to  the  center  of  the 

rocker  bearing . 
MX  =  the  bending  moment  at  "urn"  in  the  plane  of 

the  X  component  reactions. 
Hy  =  the  bending  moment  at  "mn"  in  the  plane  of 

the  Y  component  reactions. 

M  =  the  resultant  B.  M.  on  the  section, "mn" . 
f  =  maximum  fibre  stress. 

D  =  diameter  of  the  trunnions  at  section  "mn" 
I  =  the  moment  of  inertia  of  the  section  about 

diameter.  

Then  Mx=  Xa+Xrb       My=  Ya+Yrb   and  M,/^J~" 

further      nD4          MD 
I  = and  f  = 


64          21 

32M   10.18M 

hence  f  =  — —  =  r— 

nD       D 

Usually  the  fibre  strass  is  limited  from 
—  to  —  of  the  elastic  limit  of  the  material  used, 

2       3 

and    the   minimum  diarae ter   of   the    trunnions   becomes 


/10.18M 
D   =  7 

The  shear  stress  is  usually  negligible  as  compared 
with  the  bending  stress. 

LIMITATIONS  ON  THE  EXTERNAL  In  considering  the 
REACTIONS  OP  THE  TIPPING  external  reactions  on 
PARTS.  the  tipping  parts,  we 

have  to  consider  the 
limitation  imposed  on 

the  elevating  arc  reaction  and  the  reaction  on  the 
trunnions  by  the  top  carriage. 


187 


ELEVATIHG  ARC  REACTIOH: 

(1)     The  elevating  arc  reaction  is  reduced 
by  reducing  the  perpendicular  distance 
between  the  line  of  action  of  the  re- 
sistance to  recoil,  which  passes  through 
the  center  of  gravity  of  the  recoiling 
parts  parallel  to  the  axis  of  the  bore, 
and  the  trunnion  axis.   When  the  line 
of  action  of  the  resistance  to  recoil 
passes  through  the  trunnion  axis,  the 
reaction  on  the  elevating  arc  in  battery 
is  zero  if  we  neglect  the  effect  of  the 
powder  couple  and  is  equal  to  the  moment 
effect  of  the  recoiling  weight  when  the 
gun  is  out  of  battery. 

The  elevating  arc  reaction  is  re- 
duced proportionally  to  the  increase  of  the 
radius  of  the  elevating  arc. 

The  elevating  arc  reaction  should  al- 
ways be  considered  in  the  limiting  con- 
ditions of  in  and  out  of  battery,  that 
is,  with  the  maximum  powder  pressure 
couple  acting  and  out  of  battery  when 
the  maximum  moment  effect  of  the  re- 
coiling weight  about  the  trunnions  exists. 

(4)     When  the  resistance  to  recoil  does 
not  pass  through  the  trunnions  the 
elevating  arc  reaction  due  to  the  short- 
ening of  recoil  is  a  maxim  at  max.  ele- 
vation. 

TRUNNION  REACTIONS: 


(1)     The  "X"  component  of  the  trunnion 
reaction  (i.e.  the  component  parallel 
to  the  bore),  depends  upon  the  total 
resistance  to  recoil  and  the  component 
of  the  elevating  arc  reaction  parallel 


188 


to  the  bore  as  well  as  the  weight 
component  of  the  tipping  parts  when  the 
gun  elevates. 

(2)  The  "Y"  component  depends  upon  the 
weight  of  the  tipping  parts  and  the 
component  of  the  elevating  arc  pres- 
sure parallel  to  the  "y"  axis. 

(3)  As  the  gun  elevates  the  component 

of  the  elevating  arc  pressure  parallel 
to  the  "x"  axis  decreases  but  the  weight 
component  increases  and  due  to  the 
shortening  of  recoil  on  elevating  the 
resistance  to  recoil  increases. 

(4)  The  component  parallel  to  the  "y" 
axis  of  the  elevating  arc  reaction 
increases  but  in  a  negative  direction, 
thus  tending  to  decrease  the  Y  component 
of  the  trunnion  reaction.   On  high 
elevation  because  of  the  large  re- 
sistance to  recoil  for  a  short  recoil, 
the  elevating  arc  pressure  parallel 

to  the  "y"  axis  more  than  compensates 
the  decreased  weight  component  of  the 
tiooin^  parts  thus  causing  a  reversal 
of  direction  of  the  Y  component  re- 
action of  the  trunnions. 

(5)  Thus  in  general  the  X  component  in- 
creases while  the  Y  component  decreases, 
verv  often  reversing  on  elevating  the 
gun  and  thus  the  trunnion  bearing  con- 
tact may  shift  90°  or  over. 

STRESSES  IN  CRADLE  OR        The  reactions  on  the 
RECUPERATOR  FORGING.     cradle  or  recuperator  are: 
*      (1)  the  trunnion  reaction 
of  the  top  carriage  on  the 
cradle:  (2)  the  reaction 

of  the  elevating  arc  which  is  equivalent  to  a  single 
force  in  the  direction  of  the  elevating  pinion  re- 


189 


action  on  the  elevating  arc  together  with  an  addition 
al  moment:  (3)  the  reaction  of  the  recoiling  mass  on 
the  guides:  (4)  a  result  and  reaction  parallel  to  the 
longitudinal  axis  of  the  cradle  or  the  guides  due  to 
the  various  "pulls"  exerted  on  the  recoiling  mass  and 
(5)  a  distributed  load  which  is  uniform  if  the  cross 
sections  remain  the  same  due  to  the  weight  of  the 
cradle. 

In  an  accurate  computation  of  the  stresses  in  a 
cradle  it  is  necessary  from  a  preliminary  layout  of 
the  cradle  to  locate  roughly  the  neutral  axis  of  each 
section  and  connect  these  points  for  a  longitudinal 
neutral  axis  line.  "We  may  then  treat  the  cradle  as  a 
simple  beam,  talcing  into  account  the  bending  moments 
caused  by  eccentric  loads  such  as  pull  reactions  off 
the  neutral  axis,  guide  frictions,  etc.   The  trunnions 
usually  are  located  considerably  above  the  neutral  axis 
and  the  X  component  of  the  trunnion  reaction  causes  a 
large  eccentric  load  with  a  consequent  large  abrupt 
change  in  the  bending  moment  diagram.   This  is  usually 
a  characteristic  in  the  bending  moment  diagram  for  all 
cradles  or  recuperators  using  guides. 

Let  us  now  consider  the  various  diagrams  showing 
the  characteristics  for  bending  moment,  direct  stress 
and  shear  for  the  "Filloux"  cradle  as  well  as  for  the 
"240  m/m  Schneider  Howitzer"  cradle  representing  typical 
cradles  with  guides  (figures   10  and  11). 

Neglecting  the  weight  of  cradle  as  relatively 
small,  and  letting 

Mt=  max.  bending  moment  at  the  trunnions. 

Mc=  max.  bending  moment  at  the  point  of  contact 
of  the  elevating  arc  with  cradle. 

Qt  and  Q?  =  the  front  and  rear  normal  clip  re- 
actions. 

x'  and  x'  =  the  "x"  coordinates  of  these  re- 

1          2 

actions  with  respect  to  the  trunnions, 
d   and  d  =  the  distance  of  the  friction  components 

1          2 

of  Q.  and  Q  from  the  neutral  axis. 


190 


-»«- 


SENDING 


STRESS 


155  Mr^n  GUN  CRRRIRGE: 

MODEL  OP  1916  FILLOUX 
CRRDLE 


SHE1RR 


Fig.   10 


191 


B  =  the  resultant  of  the  "braking  "pulls"  reacting 

on  the  cradle. 

djj  =  the  distance  from  the  neutral  axis  to  "B" 
dx  =  the  distance  from  the  neutral  axis  to  the 

trunnions  . 

X  and  Y  =  the  trunnion  reactions  on  the  cradle. 
Xg  and  Yg  =  the  elevating  arc  reaction  on  the 

cradle  . 
Me  =  the  moment  exerted  by  the  elevating  arc  on 


the  cradle. 
=  the  distanc 


,  _^^ 

the  distance  from  the  neutral  axis  to  Xe 

xe  =  the  "x"  coordinate  of  the  elevating  arc 

contact  with  respect  to  the  trunnions. 
The  bending  moment  changes  at  the  trunnions  by 
the  amount  2X  dx. 

Now  for  guns  with  "braking  pull"  reactions  on 
the  cradle  in  the  rear  as  caused  by  the  compression 
of  the  oil  and  air  in  the  recuperator  as  in  the  155 
m/m  Filloux,  we  have  for  the  bending  moments  at  the 
trunnions, 

Mt  =  Qt(xt  +  udt)  just  to  left  of  trunnion. 
Mt  =  Qt(xt  +  udt)  -  2X  dx  (just  to  right  of  trunnion) 

and  at  the  elevating  arc 
contact, 

MC  =  V1  -  xe  *  Ud8)  -Bdb 

As  a  check,  we  also  have, 

Mc  =  Qt(x^  +  xe  +  udt)  +  2Yxe  -  2Xdx  -  Me 

The  bending  moment  MC  is  usually  distributed  caus- 
ing a  parabolic  curve  as  shown  in  the  B.  M.  diagram  of 
the  155  m/m  Filloux.. 

For  guns  with  "braking  pull"  reactions  on  the 
cradle  in  the  front,  due  to  the  tension  in  the 
stationary  hydraulic  piston  and  recuperator  rods, 
as  in  the  240  m/m  Schneider  Howitzer,  we  have  for 
the  bending  moment  at  the  trunnions 


192 


/"ft 


"1 '         »  P 

J         \ 


CRADLE 

240  MM  HOWITZER 
SCHNEIDER 
1916 


BENDlNCa  MOMENT 


DIRECT  STRESS 


SHERR 


rig.  ii 


193 


1)  -Bdj)(  just  to  left  of  trunnions)  and 
Mt=Qt  (x|  +  udt)-Bdjj-2X  dx(  just  to  right  of  trunnion) 

Further  Mc=  Qt(x|*xc+udt )+2Yxe-2Xdx-Bdb  or  as  a  check 
Mc=Gz(x^-xc+  ud,) 

In  order  to  compute  the  maximum  fibre  stress  at 
a  critical  section,  we  must  include  the  direct  stress 
caused  by  the  component  of  the  reactions  parallel  to 
the  X  axis  in  addition  to  the  fibre  stress  caused  by 
bending.  Therefore  a  direct  stress  diagram  has  been 
drawn  for  typical  gun  carriage  cradles. 

For  the  case  where  the  braking  reaction  it  in  the 
rear  from  the  front  clip  to  the  trunnion,  we  have, 

A  compression  =  uQt  which  is  small  and  nay  be 
neglected  as  compared  with  the  bending. 

From  the  trunnion  to  the  rear  clip,  we  have  at 
the  trunnions  (just  to  right  of  trunnionslsee  diagram] 

a  tension  =  X-uQ  ) 

at  the  elevating  arc  contact  section 

a  tension.  =  2X-uQ1-Xc 


Where  the  braking  reaction  is  in  front,  we  have  from 
the  front  clip  to  the  "braking  yoke"  on  the  cradle 
or  ac  a  section  through  the  point  of  application  of  the 
tensions  of  the  rods  on  the  cradle,  we  have 
a  compression  =uQ 

From  the  braking  yoke  section  the  compression 
increases  to  udt+B 
at  the  trunnions 

A  compression  =uQt+B(just  to  right  of  section) 
A  tension  =2X-uQt-B 


194 


At  the  elevating  arc  section.: 


A  ^fltftfj.,to.<ri'»3 

-ax-ua  -B-X- 

rfodffo  s   e-.-  x          c  latUio'? 

Shear  diagrams    for   the    155  ra/m    Filloux   and   240   m/m 

Schneider   Howitzer  show   the   variation  of   the   shear   in 
these    typical   cradles. 

To   recapitulate   if  yt   and  yc   =    the   distance   to   the 
extreme   fibres    from   the    neutral   axis   at    the    trunnion 
and  elevating   arc    section,    aod   if   At   and   Ac   are   tbej3£2yeo 
areas   of   these   respective   sections,    It   and   Ic  cor- 
responding  moments   of   inertia,    we   have    for    the   ex-        nn9<j 
treme   fibre   stress,  ;i,n   the  critical   sections   of   the      /sib 
ni    si   noJJsfesn   fcniJUid  erirf   si^riw   «»iso  eriJ   •;• 

qiio   Jno"jl    sdJ   fnoil   i«9i 
(    ft   =     -       ,.,  -     +  —  -     ,    ,for,toe   braking   rep  A 

action   in   the   rear.       _;  gen 

oJ  Jeuj,/ 


iaa(x+xc+uda)-Bdb]y 


or   the  braking  re- 

(       *' 

action  in  the  front. 


for  the  brak- 
ing reaction 
9w  .Jnoi'l  ai  KI  no       ln     rear 

.AiftMO 

[Q   (x   -xc+udo)]yc        uQa 

— —I — :  -        *      +    8        .,  ,  .         70 

for  the  braking  reaction 

C  AC  iu         f  i. 

in    the    front. 

.-0     fi 

iyoee   »J(OH  goi^sid  «(< J 


With  Barbette  and  Naval  mounts  the  gun  recoils 
in  a  cylindrical  sleeve  which  forms  part  of  the  cradle, 


195  aei 


. 

It  is  thus  possible  to  conveniently  locats  the  center 
line  of  trunnions  along  the  axis  of  bore  or  rather  along 
the  line  of  action  of  the  resistance  to  recoil  which  is 
usually  slightly  below  the  axis  of  the  bjpre.  Therefore 
the  elevating  reaction  is  simply  due  to  the  moment 
of  the  weight  ofowponent  of"  the  recoiling  mass  about  the 
trunnions,  though  during  the  powder  pressure  couple  we 
have  a  momentary  reaction  depending  upon  the  distance 
of  the  center  of  gravity  of  the  recqiling  mass  from  the 
axis  of  the  bore  to  the  powder  pressure  couple.   We  may 
neglect  this  effect  in  battery  as  small  and  consider  the 
reactions  out  of  battery  as  the  maximum  stress  con- 
dition. - 

In  the  design  of  -a  sleeve  cradle  it  is  highly  de- 
sirable to  locate  the  .pulls  symmetrical  with  the  axis 
of  the  bore,  this  being  completely  done  by  grouping 
the  two  separate  hydraulic  and  recuperator  braking 
systems  equally  distant  and  symmetrically  above  and  be- 
low the  axis  of  the  bore  as  in  the  16.*  model  1918  rail- 
way mount.  In  smaller  mounts  the  spring  or  recuperator 
systems  are  symmetrically  distributed  about  the  axis 
of  the  bore  and  the  hydraulic  braking  is  affected  usually 
in  a  single  cylinder  below  the  axis  of  the  bore. 

The  hydraulic  and  spring  or  recuperator  cylinders 
are  usually  clamped  in  short  distributed  bearings  to  _______ 

the  cradle  proper.   The  reaction  or  thrust  of  the  cy- 
linders is  taken  upon  shoulders  at  the  end  of  these 
bearing  contacts.   Hence  a  shoulder  bearing  surface  may 
be  regarded  as  the  point  of  application  of  the  eccentric 
load  due  to  the  pull  reaction,  on  the  cradle. 

Two  typical  stress  diagrams  for  barbette  sleeve 
cradles  are  shown  for  the  12"  barbette  mount,  model 
1917  and  for  the  16"  railway  mount,  model  1918. 

Considering  the  reactions  on  the  cradle  of  the 
.12"  barbette  carriage,  model  1917  in  figure  (12)  we 
have,  for  the  bending  moment  diagram,  from  the  front 
clip  to  the  trunnion  section,  we  have,  a  uniform  increase 
in  the  B.  M.  from   uQd   to  -• 


196 


DIRECT   STRESS 


5HERR 


BARBETTE 
MODEL.  OF  I9H 
CRRDUE. 


Hg. 


197 


In  passing  from  left  to  right  of  the  trunnion  section 
there  is  no  abrupt  change  in  B.  M.  since  the  X  re- 
action being  approximately  on  the  longitudinal  neutral 
axis  of  the  cradle  is  no  longer  an  eccentric  load. 

The  maximum  bending  moment  occurs  at  the  section 
through  the  point  of  application  of  the  eccentric 
load  due  to  the  reaction  of  the  hydraulic  cylinder 
against  the  cradle  proper. 

If  Xj!,  *  the  distance  to  this  load  d^  *  the  dis- 
tance to  the  neutral  axis,  where  Ph  *  the  hydraulic  brak- 
ing, the  bending  moment  at  this  section  becomes, 
to  left  of  section,  and 


Pndn-Qt)ud  +x'+xn)-2YxJ)  to  right  of  section. 

In  terms  of  forces  to  the  right  of  this  section  the 
B.  M.  becomes, 

phdh~Qt(xi~xh~ud)+Ye(xe~xh)*xede  to  left  of  «ection» 
and  Qt(x^-xn-ud)-Ye(Xg-xJ)-Xede  to  right  of  section. 

Since  the  elevating  reaction  is  always  very  small 
in  this  type  of  cradle,  its  influence  on  the  B.  M.  direct 
stress  and  shear  is  practically  negligible  . 

Prom  the  direct  stress  diagram,  we  note  a  maximum 
tension  (sometines  compression)  between  the  trunnions 
and  the  section  through  the  point  of  application  of  the 
eccentric  load  due  to  the  hydraulic  pall.   In  passing 
to  the  right  of  this  section  the  direct  stress  drops 
in  magnitude  equal  to  the  hydraulic  pull. 

The  shear  diagram  shows  no  reversal  of  shear  along 
the  cradle. 

Considering  the  reactions  on  the  cradle  of  the 
16"  railway  howitzer,  model  1918  (fig  13)  we  find  a. 
symmetrical  distribution  of  the  braking,  the  trunnions 
being  located  practically  along  the  axis  of  the  bore, 
thus  reducing  the  elevating  reaction  to  practically 
the  weight  effect  of  the  recoiling  mass  out  of  battery 
and  the  neutral  axis  of  the  cradle  through  the 
trunnions  and  along  the  axis  of  the  bore.  There 


19! 

} 

n 

1 

4 

«i     j. 

»-       ,JW     rr* 

v^" 

V 

i—^f-            3~         i      * 

s 

,                .     >» 

uj,. 

$ 

._  ^      ^     |_! 

~t            ~t  -~-f-^- 

I"  —  t 



1            *           * 

4     4      1 

I       i       > 

r   1 

„ 

c 

>     -*         «              * 

fl  p           ^| 

•  4     0  ^     6  5f! 

-•. 

U 

U 

E^i 

i 


•  bn* 


O\RE1CT  STRESS 


4s  ^c   /•• 

— ^^ 


• 


aoiloee 

add   0^    I*ii?a    r.  ?>   ol 

•rose   aai^Bib   it- 

^^^^  SHERR 


_  r^ 
^j 


!   &n  i  9  d 


_ 

\NCH    RRU-W/AY    HOV/\TZE1R 
MODE.U  OF 
CRADLE 


Fig.    N3 


199 


being  no  eccentric  pull  on  the  recoiling  mass  we 
have  a  distributed  load  due  to  the  weight  of  the 
recoiling  mass  as  shown  in  the  figure 

In  the  8.  M.  diagram  we  find  the  B.  M.  at 
the  trunnions  to  be  relatively  small  it  being  merely 
equal  to  that  due  to  a  distributed  load  equal  to  the 
weight  of  the  recoiling  mass,  together  with  ths 
friction  caused  by  this  loading.       w 

Thus,  the  intensity  of  loading  =  

x   +x 

*'Y3  "50  HTCH 
Hence   the  B.   N.    at   the   trunnions,    becomes, 

t  . 
jtt  Wr  x     Wr 

+   u  d 


!R*<jirt  $*u  oi  tx  xoaebqaJ 

Further  between  the  trunnions  and  the  application 
of  the  braking  to  the  cradle  by  the  brake  cylinders 
thrusting  against  the  shoulder  bearings  for  the 
cylinders  on  the  cradle,  we  have  a  tension  equal  to 
the  total  braking,  thus  causing  a  direct  stress  of 
tension  in  addition  to  the  bending. 

To  recapitulate,  for  barbette  sleeve  mounts,  if 
y=  the  distance  to  the  extreme  fibres  from  the  neutral 
axis  at  the  section  through  the  point  of  application 
of  the  maximum  eccentric  pull  due  to  the  hydraulic 
braking  or  at  the  trunnion  section  when  the  pulls  are 
symmetrically  balanced,  A  =  the  area  of  the  cross 
section  and  I  =  the  moment  of  inertia  of  the  section, 
we  have  for  the  extreme  fibre  stress 

For  barbette  sleeves  with  eccentric  pulls:— 
Section  at  eccentric  load  - 

•   ',     »x   •  leo 
(Q.  (ud+x  *xv )  +  2Yxy, )y    X— uQ 

ft=  — i ~  at  left  of  section, 

I  •>  (  AT  o  - 

and 

(P^-Q^ud+x'+XhVzYx^jy    X-Ph-uQt 
^t=  "  -   i       at  right 

of  section. 


200 


?or  barbette  sleeves  with  pulls  symmetrically 
balanced  about  the  axis  of  the  bore: 
Section  at  trunnion: 


STRENGTH  OP  CYLINDERS  AND      The  strength  of  a 
RECUPERATOR  PORGI>K3S.       recuperator  forging  is 

a  matter  of  vital  im- 
portance since  in 
modern  artillery  the 

tendency  is  to  use  higher  and  higher  pressures  con- 
sistent with  the  various  cylinder  packings  used, 
and  to  stress  the  forging  higher  with  smaller 
factors  of  safety.   Hand  in  hand  with  this  goes  the 
metallurgical  side  where  improvements  in  the  quality 
of  the  steel  with  higher  ultimate  strength  are  con- 
stantly being  made.   High  stresses  in  the  recuperator 
forgings  as  with  high  ultimate  strength  and  low 
factors  of  safety  reduce  the  weight  of  the  carriage 
and  its  cost  considerably  .  Weight  of  course  is  of 
fundamental  consideration  for  mobility.   Hence  it  is  of 
importance  to  calculate  the  stresses  in  the  cylinder 
walls  to  a  considerable  degree  of  accuracy. 

The  maxim  stress  in  a  recuperator  forging  is  a 
combination  of  the  following: 

(1)  A  bending  fibre  stress  normal  to  a 
plane  section  perpendicular  to  the 
longitudinal  axis  of  the  forging  which 

is  caused  by  the  external  reactions  exerted 
on  the  forging  during  firing. 

(2)  A  radial  compression  stress  along  a 
radius  of  the  cylinder  or  normal  to 

a  cylindrical  surface  which  is  equal  to 
the  pressure  in  the  cylinder  for  the 
inner  surface. 


201 


(3)     A  tangential  hoop  tension,  which  is 
normal  to  a  plane  passing  through  the 
longitudinal  axis  of  the  cylinder. 
Obviously  these  stresses  are  principle  stresses 
and  with  the  aid  of  Poisson's  ratio  we  may  arrive  at 
the  resultant  intensity  of  stress.   In  a  first  ap- 
proximation however  it  is  sufficient  to  consider  the 
tangential  hoop  tension  alone,  the  effect  and  magnitude 
of  the  other  stresses  small. 

Consider  now  a  single  cylinder  1"  long,  and  sub- 
ject to  an  internal  pressure  p,  and  external  pres- 
sure pt  and  of  inside  radius  R0  and  outside  radius  Rt 
respectively. 

Further  let 

r  =  the  inside  radius  to  any  differential  lamina 

of  the  cylinder  wall  (in  inches) 
dr  =  the  radial  thickness  of  the  lamina 

pr  =  the  radial  compression  at  radius  r  (Ibs/sq.in) 
pt  «  the  tangential  or  hoop  tension  (Ibs/sq.in) 

£  »  the  modulus  of  elasticity. 

Cj  =  the  longitudinal  strain. 

er  3  the  radial  strain 

et  =  the  tangential  or  hoop  strain. 

Then,  for  the  equilibrium  of  a  differential 
lamina  dr,  of  length  "1"  along  the  axis  of  the  cylinder 
and  a  peripheral  length  equal  to  the  circumference, 
we  have, 

2prlr-2(pr+dpr)l(r+dr)=2ptldr 


hence 


dPr     d(rp, 

-pp-*1  - —  » E- 

d, 


Pt  a  ~Pr-r  —  -  -  -T-^ (D 

V 

Let  us  further  assume  plane  transverse  sections 
to  remain  plane  under  pressure.  This  assumption  is 
reasonably  close  to  actual  conditions  for  plane  trans- 
verse sections  some  distance  from  closed  ends,  and  in 
the  case  of  a  recuperator  forging  where  the  intensity 


202 

of  longitudinal  stress,  i.  e.  the  bending  stress  on 
transverse  sections,  is  relatively  small,  except  for 
extreme  fibres  from  the  neutral  axis  of  the  transverse 
section. 

We  are,  therefore,  not  greatly  in  error  in  as- 
suming the  longitudinal  strain  to  remain  constant 
over  the  entire  cross  section,  hence, 

vswcd  c.: 

ej   =  -   (pi )   =  constant^1    q°< 

.  ii  ;   edJ    lo 

where   p^   =    intensity  of   longitudinal  stress, 

— ev  nt    oi    Jost 

hence   pt-pr=k  (2)  ^    MU( 

dPr  dP 

Pt"Pr=   ~2Pr"r  cT^—  or  k   +   2pr=  -r  ^7- 

therefore  dpr       ^ 

k+2Pr  "     ~ 

Integrating,  log(k+2pr)=  -  leg  r*  +  c  or  k+2pr=  — - 

r 

c    k 
hence  pr  =  — -  -  -  (3) 

^  r*     & 

^c_   k  «*•»«•  qoc 

2r*    2 

, 

Substituting  (3)  in  (2)  where  pr=po 

r  =RQ   inside  conditions 


r  =  Rt  outside  conditions 

c     k 
hence  P Q- 

po   2 

. 

c    k 

P  =    '  -  -  oJ 
1   2R2    2 
i_ 


p  _p   _  _  fK\  -39  88T9V 

ro  ri  vo^ 

,T  a  to 


203 


Now  eliminating  c  and  k,  respectively,  we  find 


t     „ 

c  -  --  -  -    and  k  = 


r    i    <i  -3*1   )HoH  I1        j3*      ~"o3o     S 
Substituting   in   (5)  we  have, 


—      - -   _         I     _L^^^_J-_.-»_  T-.    V_JT_«^«—       

r  RZ-R2         R2-B8        r2    (6)    (  Apparent 

(  stress. 


o  . 


p   =  , — +  _^ —  (7)          ( 

HJ-B*  Rt2-R«  r»  )b,   Uiq»i« 

The  stress  corresponding  to  the  actual  strain 
produced  in  the  material,  which  is  the  basis  of  stress 
limitation  imposed, (assuming  m  =  3  Poisson's  ratio), 

becomes, 

.noiaoeJ    qoo/i   ecU   n*rfJ    **el    «x*»i*   •*   ool»e»Tq»oc   itJt-8T 

Eet=St=E(^+~  -  £i)  (8)  £i   Actual 

£    m£        raE 

(   stress 

)   corres- 
p,.        p.        P]  (   ponding 

n  o  f  f      r  *  *\  /n\          ^891^8      «U-J  , 

Eer=S_=  -E(—  +   —  +  — )          (9)  )    to   actual 

(   strain 

Where    (6)    and    (7)    are   substituted    in    (8)   ami    (9)    and 

o2 

i-i-ii.     p  ° 

F^-RQ 

Jotet«<4    bn*    {        .^f'J) 

the  above  expression  reduces  to  Clavarine's  formula. 
Birnie's  formula  is  a  modification  of  the  above 
assuming  p^  =  0,  that  is  no  external  longitudinal 
tension.   Usually  pi  is  relatively  small  and  hence 

•*  A 

(8)   and    (9)    simplifying    to,      Uw   1<,01»J*< 
eriJt   lo    lifiw  co  -    ebiaJyo  «dJ    ^o   *noiap9^   oocti   sd* 


204 


Eet  -  S  -  -  —  +  =  -3-2 i-  —       (10) 

*     3  R«-Rt  3  R»-Rt  r» 


rg  3  R.-  R, 

The  maximum  hoop  tension,  therefore  becomes, 
2R«+4R« 


which  gives  slightly  higher  values  than  Lane's  when 
simplified,  that  is, 

Rg  *  R; 

St*  P  o«-R2  Lame's  foraula    (13) 

Kt  Ro 

From  the  above  formulae  it  is  evident  that  the 
radial  compression  is  always  less  than  the  hoop  tension. 

With  large  bending  fibre  stress  due  to  external 
reactions  on  the  recuperator  forging, 

pj  is  no  longer  equal  to  zero 

The  maximum  stress  which  is  the  hoop  tension  becomes, 

(14) 
max 

THICKNESS  OF  WALLS  BETWEEN     Considering  two  parallel 
ADJACENT  CYLINDERS.        cylinders  bored  in  one  forg- 

ing (fig.   )  and  passing 
a  longitudinal  plane  section 
through  the  center  lines  or 

axes  of  cylinders  (1)  and  (2),  we  have  either  half  of 
the  forging  above  or  below  this  plane  section  in 
equilibrium  under  the  internal  hydrostatic  pressures 
(which  now  are  external  with  respect  either  half)  and 
the  hooo  tensions  of  the  outside  and  common  wall  of  the 


205 


two  cylinders.   Further  if  the  two  cylinders  are 
under  pressure  pt  and  p  respectively  and  neglect- 
ing the  small  variation  of  the  hoop  tension  for  dif- 
ferent radius,  we  have  for  a  close  approximation 

ptCT,  +  T,  +  «)  -  ptdt  +  ptdf 

where  Tt  and  Tf  »  thickness  of  cylinder  walls  (1) 

and  (2)  respectively. 
w  »  total  width  of  common  wall  between  the  two 

cylinders  . 
dt  and  df=  the  diameters  of  the  respective 

cylinders. 
pt  =  the  assumed  allowable  hoop  tension  fibre 

stress. 
Simplifying, 


Pt 

For  a  correction  due  to  the  fact  that  the  hoop 
tension  is  not  constant  but  varies  sligatly  with 
the  radius,  we  nay  augment  w  by  decreasing  pt  to  0.9 

Pf 

Further  due  to  symmetry 

Pd  P2da 

and  ^  "  ~* 

and  substituting  in  the  previous  equation,  we  have 


w  = 


1-8  Pt 

which  gives  the  minimum  thickness  of  wall  between 
two  cylinders  under  pressures  pt  and  p8  respectively. 
Evidently  the  maximum  simultaneous  pressures  in  the 
two  cylinders  should  be  considered  together. 


206 


ALLOWABLE  STRESSES  IN       Though  this  matter  will  be 
CYLINDER  WALLS.         taken  up  in  detail  in  practical 

design  applications,  certain 
limitations  could  profitably 
be  mentioned  here. 

Cylinders  should  be  tested  for  strength  at  pressures 
considerably  higher  than  would  be  used  in  service.   It 
is  imperative  that  even  under  test  pressure  the  elestic 
limit  is  not  exceeded.   Test  pressures  should  be  at 
least  1  —  and  preferably  twice  the  maximum  service 

pressures  and  these  test  pressures  should  not  exceed 
3  the  elastic  limit  of  the  material  or  4_ proportional 


limit. 


TOP  CARRIAGE  The  forces  exerted  on  the 

top  carriage  are  the  reactions 
of  the  tipping  parts  and  the 
supporting  forces  of  the  plat- 
form, or  bottom  carriage,  or 

ground  and  axle  for  a  trail  carriage.   The  reaction 
exerted  by  the  tipping  parts  on  the  top  carriage  may  be 
divided  inter- 
CD     The  trunnion  reaction. 
(2)     The  reaction  of  the  elevating  arc  of 
the  tipping  parts  on  the  pinion  of  the 
top  carriage. 

The  tension  of  the  equilibrator  chain 
or  rod  where  an  equilibrator  is  used. 
These  reactions  are  balanced  by  the  supporting 
forces  exerted  at  the  base  of  the  carriage. 

In  figure  (14)  the  reactions  on  the  top  carriage 
are  shown. 

Considering  now  the  reaction  of  the  tipping  parts 
on  the  top  carriage  assuming  that  an  equilibrator  is 
not  used.   Taking  moments  about  the  hinge  point  A(as  in 
previous  discussions),  we  have,  when  the  gun  has  recoiled 
a  distance  X  out  of  battery. 


207 


1 


noiJBupe  svcds  sri-i  ni  ssui^v 


i«J  ,§• 


-iR.  14- 


.III  TsJqsriO  aaS      fa  -  2+S  nie^l-^  aoo^ri  woM 


:ods  1H  'jo  BT6 


208 


2X(ht-it  sio  0)-2Y(lt  cos  0+ht  sin  0) 

-  E  cos(ee-  0)[ht-j  cos(9e-0)]-E  sin(ee-0[lt-j  sin(ee-9)] 


'ta 


Mo"  (Kps+Wpx  cos  0) 

2X»Kr+Wt  sin  0+  • cos  9( 

J 

(Krs+Wrx  cos  0) 
2Y=Wtcos  0-       .         sin  9e 

and      Krs+WrX  cos  0 

in   _ 


J 


Substituting  these  values  io  the  above  equation, 

Krs+WrX  cos  0 

r+Wtsin  0+  ( •  •   )cos  9e](htcos  0-ltsin 

J 

Krs+WrX  cos  0 

-!Wtcos  6-( : )  sin  6e](lt+  cos  0+btsin 

J 


[KP+Wtsin  0+  ( ,       )cos  9e](htcos  0-ltsin  0) 

0) 
J 

Kps+WrX  cos  0 


cos(ee~  0)(ht-j  cos(9e-0)l 


-(Krs+WrX  cos)sin(9e-0)[lt-j  siD( 

Expanding  and  simplifying,  the  above  reduces  to: 

Kr(htcos  0-ltsin  0+S)-wtlt+»rX  cos  »  Mta       (1) 

How  htcos  0-ltsin  0+S  =  d      See  Chapter  III. 

where  d  is  the  moment  arm  of  Kr  about  the  hinge  point  A. 


209 


Further  due  to  the  displacement  of  the  recoiling 
parts  a  distance  X  from  the  battery  position,  the 
center  of  gravity  of  the  tipping  parts  is  displaced 
a  distance  ffrcos  ^x     from  the  initial  trunnion 

Wt 
position. 

The  moment  of  the  weight  of  the  tipping  parts 
about  A,  is  therefore, 


r 

Wt(lt--r-X  cos  0)  =  *tlt-Wrx  cos  0 
™t 

Hence  from  equation  (1)  we  observe  that  the  reaction  on 
the  top  carriage  is  equivalent  to  the  total  resistance 

to  recoil  through  the  center  of  gravity  of  the  recoiling 
parts  together  with  the  weight  of  the  tipping  parts  act- 
ing at  a  distance  IE  x  CQS  ^  from  the  trunnions.  There 

fore  the  line  of  action,  is  equivalent  in  effect  to 
the  resultant  of  the  trunnion  and  elevating  arc  reaction. 
This  is  almost  obvious  from  first  principles  since  the 
reaction  of  the  top  carriage  on  the  tipping  carts  must 
balance  the  resultant  of  Kr  and  Wt:  hence  by  the  law  of 
action  and  reaction,  the  resultant  reaction  of  the 
tipping  parts  on  the  top  carriage  is  therefore  equal  in 
magnitude  and  direction  to  the  resultant  of  Kr  and  Wt. 

With  a  balancing  gear  we  have  in  addition  to  the 
trunnion  and  elevating  arc  reaction  on  the  top  car- 
riage, (which  now  have  different  values  from  the  pro- 
ceeding) the  tension  of  the  equilibrator  chain  or  rod. 

By  exactly  a  similar  analysis  as  in  the  above,  the 
reaction  on  the  top  carriage  reduces  to  the  resultant 

of  Kr  and  Wt  where  the  line  of  action  of  the  component 

• 

Wt  if  now  disolaced  a  horizontal  distance  __r_ 

^  X  cos  /)  - 

the  distance  which  the  center  of  gravity  of  the  tipping 
parts  in  battery  is  placed  backwards  from  the  trunnion 
position,  when  the  balancing  gear  is  used. 


210 


In  figure  (14)  is  shown  the  various  reactions  on 
the  top  carriage  together  with  a  force  polygon.  Thus 
from  the  space  diagrams  of  reactions  obviously  the 

lines  of  action  of  the  resultant  of  the  trunnion  re- 

?, 
action  and  elevating  arc  reaction  intersect  at  a  com- 

mon point  which  necessarily  lies  along  the  line  of 
action  of  the  resultant  of  Kr  and  Wt  where  the  com- 
ponent of  Wt  is  displaced  a  horizontal  distance 


X  cos  <6  from  the  trunnion  axis. 

* 

In  the  vector  polygon  of  forces  we  note  that  by 
vector  addition,  K+»t-X+T+E 

Further  for  the  equilibrium  of  the  top  carriage, 
X+Y+E+Ha+Va+Vb  =  0   hence  K+Wt+Ha+Va+Vb  =  0 
The  above  results  are  exceedingly  valuable  in 
graphical  methods  as  will  be  outlined  later  for  ob- 
taining the  various  reactions  throughout  a  gun  car- 

riage. 

on  ic-.  ;pe  «i  .noijse  1o  enrl  edi  eiol 

•if-  fioifmuiJ  arfo 
SUPPORTING  REACTIONS       Top  carriages  have  been 

OW  VARIOUS  TYPES  OF   q  classified  in  Chapter  I,  into 
TOP  CARRIAGES.          l[l)  the  ordinary  type  with 

side  frames  and  connected  at  front 
or  rear  by  cross  beams  or  trans- 

oms, which  contain  the  pivot  bearing,  (2)  pivot  yoke  type 
used  on  small  mobile  mounts  and  (3)  trail  carriage. 
The  supporting  reactions  in  the  ordinary  type  of 

top  carriage  are  the  H  and  V  components  of  the  pivot 

-•  v  ct  ri  Kd/i  H  **>  riiwi  fi^£.f*f 
bearing  which  is  usually  in  the  front  and  the  V  com- 

ponent exerted  by  the  traversing  circular  guides  in  the 

rear.   Sufficient  horizontal  play  is  allowed  so  that  the 

• 
reaction  of  the  horizontal  traversing  guides  is  only 

• 

vertical,  the  H  component  being  taken  up  entirely  at  the 
pivot  bearing. 

As  a  typical  class  (1)  top  carriage  we  may  illus- 
trate by  the  Vickers  8",  Mark  VII,  British  Howitzer. 

;  siU  a-oil  --SJ.TBCJ  ni  EJieq 


211 


Further  let 

1  =  distance  between  supporting  reactions  measured 

horizontally  in  the  direction  of  the  axis 

of  the  bore  at  0°  traverse. 
A  =  the  front  pivot  point. 

•  r 

B  =  the  resultant  of  the  distributed  vertical  re- 
actions of  the  horizontal  traversing  arc  guide. 
1  k  =  the  horizontal  distance  to  trunnions  from 

B  in  the  direction  of  the  axis  of  the  bore 

o 
at  0   traverse. 

ht  =  height  of  trunnions  above  the  traversing 
guides. 

S  =  the  perpendicular  distance  from  the  trunnion 
center  to  the  line  of  action  of  the  resistance 
to  recoil  which  necessarily  passes  through 
e  center  of  gravity  of  the  recoiling  mass. 
=  height  of  horizontal  component  of  pivot  re- 
action above  the  horizontal  traversing  guides. 
=  weight  of  top  carriage  proper. 
=  moment  arm  of  W   about  B. 


Considering  fig. (15)  we  have  for  the  horizontal 
component  of  the  pivot  reaction,  Ha=K  cos  /5 
and  taking  moments  about  fl,  the  center  of  pressure 
of  the  traversing  guides,  we  have, 
Kd-W^-t*Wrx'cos  0-Wtcltc+Val-Haha  =  0 

(  . 

Wtlt+Wtcltc-Wrx  cos  0-K(d-ha  cos  0) 
)  hence  Va=  •  '   ' 


*-ha  cos  0)+Wt(l-lt)+lftc(l-ltc)-»-H'rx  cos  0 
(  and  Vb  =  


where  for  low  angles  of  elevation,  d=htcos  0+S-ltsin 

d'  =  htcos  ^-«-(l-lt)sin  0+S 


212 


Fig.  15 


213 


and  for  high  angles  of  elevation, 
d*ltsin  0-btcosl  0-S 

d'  =  (l-lt)  sin  0+htcos  0+S 

ymn  =  the  horizontal  distance  along  the  axle  from 
the  center  of  the  wheel  bearing  pressure. 

Considering,  max.  traverse,  right  handed,  at 
max.  elevation,  the  reactions  on  the  axle  to  the 
left  of  the  section,  become, 

(1)  The  components  of  the  trail  con- 
necting arm  reaction  on  the  axle:- 
X,Y  and  Z  together  with  a  couple  Mxy 

in  the  horizontal  plane. 

(2)  The  vertical  reaction  exerted  by  the 

left  wheel,  Sa.   Therefore  at  section 
"mn",  we  have, 

(1)     Bending  in  the  vertical 
plane: 


(2)     Bending  in  the  hor- 
izontal  plane: 

<in  lbs-> 


(3)  Shear  in  the  vertical 
plane  : 

X+Sa  (Ibs) 

(4)  Shear  in  the  hor- 
izontal plane: 

X  (Ibs) 

(5)  Torsion  about  the  y 
axis,  or  in  the  x  z 
plane:  T=X  ZBn 

(6)  A  direct  thrust: 

Y  (Ibs) 

Thus,  we  have  bending  in  two  planes  combined  with 
torsion,  and  a  direct  thrust  as  well.   Then  for  a 


214 


round   section,    as   would^  Dually  be   the  case,   we  h*»»,jon6 


f   3  -    -  +  -  _  Max%    normal   fibre 

°-78J  thrust   on  outer   layer 

•oil    »fx*   etu  c,ns.t8ifa   1*5  (Ibs/sq.in) 

gniised   Jaedw   &dJ    lo  -iscfnso   sr)J 

,  .- 
Oj   elxfi   eri^   no 


iis-U    od^   io 

-ralxs   ori^    no   noj  ";s   iniJoso 

The   m-axi/num  fibre    stress,    therefore,   becomes 

_  ._an£tq   Ifi-tnoxiiori    erid    ni 


f   =-f   +  /¥**"£  Jo*  en    ]80lj18v  ftbs/sq.in) 

3         4 

•  a  2  *  j  • 

2 

which  should  not  exceed  —  of.  the  elastic  limit  of  the 
material  to  be  used. 

As  a  typical  class  o'f  pivot  yoke  type,  consider 
the  reactions  on  a  Barbette  or  Pedestal  mount,  figure 
(16)  and  a  pivot  yo.ke  top  carriage  used  on  a  trail 
carriage,  fig.ure  (16A).   In  the  first  type,  the  lower 
bearing  sustains  both  horizontal  and  vertical  com- 
ponent reactions,  whereas  the  upper  is  merely  a 
floating  bearing  and  therefore  sustaining  only  a  hor- 
izontal component  and  designed  to  prevent  bending  in 
the  lower  pivot. 

In  the  second  type,  the  middle  bearing  has  a 
tapered  fit  within  the  axle,  and  therefore  sustains 
both  horizontal  and  vertical  components,  but  suffers 
no  bending  moment  since  the  axle  is  free  to  rotate. 
To  prevent  the  top  carriage  and  mount  from  rotating  about 
the  axle  a  lower  cylindrical  vertical  pivot  fits  within 
an  equalizer  bar  below  the.  axle,  the  equalizer  bar 
being  attached  to  the  trails.   (See  Theory  of  Split 
Trail  -  next  section). 

In  this  type  of  mount  it  is  more  convenient  to 
compute  the  supporting  reactions  in  terms  of  the  hor- 
6  10^  nsriT   .Ilsw  8£  Jsind.*  joetifa  £  b.~. 


215 


foul  as   o 


l«3JUi»tr   bn*   isJ-nosx 


aoioiq  Sai<JBV8is  04    z^ibsi   V.Q  absa 
-lev   isdJ    diiw  jip*i    2oiJ*voI»  no 

REftCT\ONS  ON 
P\VOT    >TOKE.   TOP 


cerfT 


1 J    II 

^4^ 


IS    8OO 


~n   ale    {- 


aie 


:®n — -ZH 


216 


izorual  and  vertical  components  of  the  trunnion. 

If 

ne  =  angle  made  by  radius  to  elevating  pinion 
contact  on  elevating  rack  with  the  ver- 
tical. 

j  =  radius  of  elevating  rack 

Then  for  the  horizontal  and  vertical  components  of 
the  trunnion  reaction,  we  have, 


Fe+Ks 

>K  cos  0+ ( )  cos  ne  )   In  battery 

j  ( 

Fe+Ks  ) 

<K  sin  0+Wt-(— : )  sin  ne  ( 

J  ) 


and 

Ks+Wrb  cos  0 
2H  =K  cos  0+  ( )  cos  ns    )   Out  of 

J  (   battery. 

Ks+W_b  cos  0 
2V=K  sin  0+Wt-( )  sin  ne   ( 


For  the  elevating  gear  reaction,  we  have 

Fe  +  Ks 

in  battery 

Ks+Wrb  cos  0 


j 


out  of  battery 


In  the  Barbette  or  Pedestal  Mount,  figure  (16) 
let, 

xt  =  distance  from  center  line  of  pivot  to 

center  of  trunnions. 
yt  =  height  of  center  of  trunnion  from  bottom 

of  rivot. 
T  "  radius  of  pivot  bearing. 


217 


r*f  =  radius  of  floating  bearing. 

y^  =  height  between  bottom  of  pivot  and  top 

of  floating  bearing. 
Then,  Va  =  2V  +  E  sin  ne 

and      1   ( 

Hb=  ~T  (  2Hyt+2V(xt-rp)+Etj+(xt-rp)sin  ne-yt  cos  ne] 
"m 

and  Ha=Hb+E  cos  ng-2H 

In  the  pivot  yoke  trail  top  carriage,  fig.(16A),  let 
x^  =  distance  from  center  line  of  pivot  bo  center 

line  of  trunnions 
yt  =  distance  from  center  of  axle  to  center  of 

trunnions  . 
ym  =  distance  from  center  of  axle  to  center  of 

equalizer  beam. 
Then,  Va=2V+E  sin  ne 

and  H(j=  -  [2Hyt  +  2Vxb+E(o+xtsin  ne-ytcos  ne)] 


Ha=Hb+2H-E  cos  ne 

THEORY  OF  SPLIT  TRAIL.      The  object  of  a  split  is 

primarily  to  give  a  large 
aperture  between  trails  for 
the  gun  to  recoil  at  maximum 
elevation  and  maximum  traverse 

When  split  trails  are  used  it  is  also  desirable  to  dis 
tribute  the  bearing  load  on  spades  when  the  gun  shoots 
at  maximum  elevation.  T"his  is  accomplished  by  the 

use  of  an  equalizer  bar  connecting  the  two  trails,  or 
more  strictly  the  trail  arms,  beneath  the  axle,  the 
equalizer  laying  usually  in  a  horizontal  plane  and 
pivoted  at  its  center  by  a  vertical  pin  through  the 
center  of  the  axle.   With  split  trail  and  equalizer, 
a  pivot  yoke  type  of  top  carriage  s-hould  be  used. 
The  elements  of  a  slip  trail  mechanism  are  :- 
(1)     The  two  trails  with  their  spades 
which  are  connected  by  a  vertical  pin 


218 


n  o  I  i  s  v 

5  7  '   & 

2|  *»  KT^ 

I        ^  x  i... 

/ 

toannoojifid 
"^  "V     i 

i 


n 


219 


to  two  trail  arms  or  trail  connect- 
ing pieces,  at  either  end  of  the  axle. 

(2)  The  trail  arn  or  connecting  pieces 

are  free  to  turn  about  the  axle  in  a  ver- 
tical plane  and  are  prevented  from  slid- 
ing along  the  axle  by  thrust  shoulders. 
The  moment  about  the  axle  of  the  trail 
reaction  is  balanced  by  the  moment 
about  the  axle  of  the  shear  reaction 
,,£t  the  equalizer  bar. 

(3)  The  equalizer  bar  is  usually  de- 
signed to  rotate  in  a  horizontal 
plane  about  a  vertical  pin  through 
the  axle.  Thus,  with  a  split  trail 

we  have  the  two  trails,  their  connect- 
ing pieces  to  the  axle  and  equalizer 
and  the  equalizer  bar,  connecting 
the  trail  arms  and  pivoted  about  a  ver- 
tical pin  which  passes  through  the 
center  of  the  axle. 
We  have  the  following  possible  motions:- 

(1)  A  free  rotation  in  a  horizontal 
plane  of  either  trail,  about  the 
vertical  pin  in  the  trail  arm. 

(2)  A  constrained  rotation  in  a  vertical 
plane  about  the  axle  of  either  spade, 

the  constraint  being  due  to  the  equalizer 
bar. 

(3)  A  constrained  rotation  in  a 

horizontal  plane  of  the  equalizer  bar 
about  a  vertical  pin  through  the  axle. 


MAXIMUM  BLBVATION: 


Let  x  and  y  =  horizontal  coordinates  in  longitudinal 
and  transverse  directions  respectively, 


220 


i.  e.  along  and  cross  wise  to  the  bore 

at  zero  traverse. 
Z  -  vertical  coordinate. 
AXAVAZ  =  the  component  reactions  at  the  left 

spade  (positive  direction  towards 

muzzle  (Ibs) 
8xByBz  =  the  component  reactions  at  the  right 

spade  (Ibs) 
Sa  and  Sb  =  normal  vertical  reactions  for  left 

and  right  wheel  respectively.  (Ibs) 
Qa  and  QJJ  =  shear  reactions  of  equalizer  on 

trails  A  and  B  respectively    (Ibs) 
dj,  =  horizontal  distance  or  span  of  equalizer 

between  trails  which  it  connects  (in) 
de  =  vertical  distance  from  center  line  of 

equalizer  to  center  line  of  axle  through 

wheel  hub.    (in) 
£Mav  =  moments  of  the  components  of  A  about  the 

axle.    (in  Ibs) 
SMQV  =  moments  of  the  components  of  B  about  the 

axle  .   (in  Ibs) 

jc_  =  distance  from  spade  to  axle.   (in) 
yo  =  distance  from  ground  to  center  line  of  axle. 

(in) 
2Ma[,  =  moments  of  reactions  of  A  about  vertical 

pin  in  left  trail  am. 
SMfj  3  noments  of  reaction  of  B  about  vertical  pin 

in  right  trail  arm. 
Taking  moments  about  the  center  pin  of  the  equalizer, 

we  have, 

Qa  111  =  Q.  Ik. 
a  2    b   2    hence  Qa  =  Qb  =  Q.  Therefore, 

for  moments  about  the  axle,  we  have 


-Q  de  =  0 


hence,  2Ma  *  2Mb 


We  have  for  unknowns, 


Sa    Sb 


221 


(  Sight  unknowns 


Equations  for  solution: 


ZX 


£M 


ZY=  0 


ZMy  =0 


X  =  0 


=  0 


2Mbh=  0 


) 


)  Nine  solutions 

( 

) 

( 

) 


We  therefore  would  expect  either  2Mah  or  ZM^j,   not  zero 

This  is  physically  justified  since  on  extreme  traverse 
one  of  the  wheels  and  trails  must  be  in  contact.  This 
is  met  constructively  by  usually  introducing  a  show 
attached  to  the  trail  which  comes  in  contact  with  the 
wheel  upon  traversing. 

If  N  =  normal  reaction  between  shoe  and  wheel, 

dD  =  perpendicular  distance  from  vertical  pin 

on  trail  arm  to  vertical  plane  through  wheel, 
i.  e.  to  line  of  action  of  N. 
We  have  for  maximum  traverse  in  a  right  banded  rotation, 

^bh=^  ^n  *nus  introducing  an  additional  unknown 
N.   The  solution  is,  therefore,  statically  possible 
either  introducing  the  above  equation  or  omitting  it 
entirely. 


METHOD  OF  SOLUTION 

Assume  maximum  traverse  right  handed  turn, 
Let  0ffl  =  maximum  angle  of  elevation. 
8m  =  maximum  angle  of  traverse. 


222 


Ks  =  resistance  to  recoil  at  maximum  elevation 

,,.  .  :v*ri  eW 

(Ibs) 

Xy  =  horizontal  distance  to  projection  of  center 
of  gravity  of  recoiling  parts  measured  from 
base  line  AB.  (in) 

yg  =  0  assumed  approx. 

Zg  =  height  of  center  of  gravity  in  battery  above 

ground,    (in) 
g  =  vertical  distance  from  ground  line  to  center 

of  pressure  on  spade  .  (in) 
Wg  =  weight  of  total  system,  gun  +  carriage.  (Ibs) 


g 

=  horizontal  distance  from  AB  to  W 


,, 

Li  *7 


y  A'B1  =  distance  between  vertical  trail  pins. 
Then,  the  resolved  component  of  Ks  through  the  center 
of  gravity  of  the  recoiling  parts,  become, 

/<MS 
Kscos  0m  cos  em,  along  the  x  axis      ) 

Kscos  6^  sin  Q  ,  along  the  y  axis      ) 

(  r.  i  ri  T 

K  sin  <L         along  the  Z  axis      ) 


xp  =  distance  from  AB  to  either  vertical  trail  pin. 

yab  =  distance  between  A  and  B,  trails  completely 
,    neswJed  tv         i-ioa  =  n  11 


, 
spread. 

Taking  moments  about  A  B,  we  have, 


:  muaix 
hence,       W  1  -Kscos  0  cos  9m(Z.+  g)+Kssin 


•  x.  -  -, 

.pa  «vo4s  art*  r-  .:is 

and  Az+  Bz=  V  Ks  sin  ^«-(Sa+  Sb  )  (2) 

Next  take  moments  in  a  horizontal  plane  about  the  left 
spade,  and  we  have, 

.  .t  £  v  a  1  s  J^ab 

= 


. 

em.  xg-Kscos  Bf.cos  9m  -j-  =  0 


223 


K  cos  0m 

hence  Bx  =  -  (0.5  yab  cos  9n~Xg  sin  6m)    (3) 
yab 

Further  Ax+Bx=K3cos  J0m  cos  0m 


hencs  AX=KS  cos  Am  cos  9m-Bx     (Ibs)  (4) 

For  moments  about  the  vertical  pin  in  trail  arm  for 
left  trail,  we  have,   Ayx_-Ax  0.5  y/^'B1  ~  0 

0.5  AxyAnj» 
hencs  Au  =  -         (Ibs)  (5') 

XP 
Now  if  we  take  moments  about  the  axle,  we  have  ZMgy^ 

Az   *0-W*>  =  Bzxo-Bx^o^> 
therefors     (A-B  )  (Z+g) 


'Z  "Z      X, 


but 


lience  Az   =~       2x  ~~^~  (6) 

Bz=W>»+Kssin   e)m-(Sa+Sb)   -  Az  (7) 

Let 

X,  Y  and  Z  =  components  of  the  reaction  of  the 

trail  arm  on  the  axle, 
MXV  =  moment  reaction  of  trail  arm  on  axle,  in 

the  X  Y  plane. 

Considering  moments  on  the  left  trail  and  trail 
arm  together  about  the  axle,  we  have, 

AzxQ-Ax(Z0+g)-Q  de  =  0   hence,  the  horizontal 

shear  reaction  of  the  equalizer  becomes, 


*. 


Next  consider  the  various  reactions  on  the  trail  arm, 
and  we  have, 


224 


=  °  along  the  x  axis, 
-Y+A  =  0  along  the  y  axis. 
-Z+AZ  =»  0  along  the  z  axis. 

and  further,  -Mxy+Ay(xo-xp)  '0   In  the  x  y  plane 
Therefore,  the  reactions  of  trail  arm  on  the  axle, 

becomes,     A_xo-Ax(Z0+*) 

X=A+A  »   z  o   *   o +  A.,    (Ibs)     (9) 

do 


Y  =  Ay  (Ibs)     (10) 

Z  «  A,  (Ibs)     (11) 


Mxy»  Ay(x0-xp)  (in. Ibs)   (12) 


Of  AXLS  MAXIMUM  TRAVBR8B,   MAXIMUM  ELEVATION: 

This  critical  section  of  an  axle  is  at  a  section 
near  the  center  where  the  axle  becomes  enlarged  for 
holding  the  vertical  pivot  of  the  top  carriage.   If 
the  axle  is  made  straight,  we  have  no  torsion  on  the 
section  but  mersly  bending  in  a  vertical  and 
horizontal  plane.   If,  however,  the  axle  is  underhung 
for  clearance  and  lowering  the  top  carriage,  in 
addition  to  the  bending,  we  have  torsion  as  well,  the 
nagnitude  of  the  torsion  depending  upon  the  depth  of 
the  underhang. 

Let  mn  be  the  section  under  consideration  near 
the  center  of  the  axle. 

xnn  ^mn  an^  zmn  =  ^9  component  distances  from 
the  center  of  contact  of  the 
trail  connecting  arm  and 
axle. 

REACTION  BETWEEN  RECOILING  During  the  counter  re- 
PARTS  AND  MOUNT  IN  COUNTER  coil,  we  may  distinguish 
RECOIL.  between  the  accelerating 

and  retardation  period  so  far  as  the.  reactions  between 


225 


the  recoiling  parts  and  mount  are  concerned.   The  re- 
actions during  the  acceleration  however  are  of  exactly 
the  same  character  as  during  the  recoil  only  of  less 
magnitude.   Therefore,  from  either  the  point  of  view 
of  the  internal  reactions  or  stability  of  the  mount, 
we  are  not  concerned  with  the  acceleration  period  of 
counter  recoil. 

Therefore  let  us  consider  the  various  recoiling 
parts  and  mount  coming  into  play  during  the  retardation 
period  of  counter  recoil, - 

Let       (see  figure  18) 

xt  and  vt  =  coordinates,  along  and  normal  to 
bore,  of  front  clip  reaction  with 
respect  to  center  of  gravity  of 
recoiling  parts. 

x   and  y   =  coordinates,  along  and  normal  to 
bore,  of  rear  clip  reaction  with. 
respect  to  center  of  gravity  of 
recoiling  parts. 
Qt  s  front  clip  reaction. 
Q2  =  rear  clip  reaction. 
Wr  =  weight  of  recoiling  parts. 
0  =  unbalanced  retarding  force  exclusive  of 
f rict  ion. 

^0  -  distance  from  center  of  gravity  of  re- 
coiling parts  to  line  of  action  of  0. 

n  =  coefficient  of  friction  =  0.15  usually. 
d1  =  distance  from  front  wheel  ground  contact 

to  line  parallel  to  tore  through  center 

of  gravity  of  recoiling  parts. 
lr=  horizontal  distance  from  line  of  action 

of  Wr  to  front  wheel  ground  contact. 
x  =  displacement  along  bore  or  guides  from 

out  of  battery  position. 
Ma  =  moment  of  reaction  of  the  recoiling  parts 

on  mount  about  front  wheel  contact  and 

ground. 
Then,  for  the  motion  of  the  recoiling  parts,  we  hava. 


226 


REACTION  BETWEEN  RECOILING  PRRT5 
RND  MOUNT  IN  COUNTER  RECOIL 


RECOILING    PRRT5 


227 


d  x 

0+n(Qt+Q2)+Wrsin  0=-mr  — -  (1) 

d  t 


Qt-Q2=Wrcos  0  (2) 

and  0  d0-ftlx1-Qfx8+n  Qtyt-n  Q8y2  =  0  (3) 

Next,  considering  the  reactions  on  the  mount  and  taking 
moments,  about  the  point  of  contact  of  the  front  wheels 
with  ground  A  ,  we  have, 


(4) 


-  -  d'tan  0+x  J=MrA' 
0 


Substituting  Eq.  (3)  and  (2)  in  Eq.  4,  we  have 
immediately 


d"x 


rQ    )d   +Wrsin   0.d  -W_l   = 

*          2 

or 

(0+n(Qt+Q2)+Wrsin  0)d '-Wrlr=MA 

Further   from   equation    (1)        (-mr        ^    )d'-  Wrlr=  My 

Q  t» 

Hence,  the  reaction  on  the  mount  during  counter  recoil 
is  equivalent  to  the  total  resistance  to  recoil  acting 
in  a  line  parallel  to  the  axis  of  the  bore  and  through 
the  center  of  gravity  of  the  recoiling  parts,  together 
with  a  component  in  line  and  equal  to  the  weight  of  the 
recoiling  parts. 

If  further,  we  let, 

Fy  =  the  recuperator  reaction. 

RQ  =  total  guide  friction 
RS+P  3  total  packing  friction. 

Bx  =  total  counter  recoil  buffer  reaction. 
Then  0  =  B '+£„..  n  -  F,, 


aa  )  =  n  Wr  Cos  0   (approx.) 
and  the  overturning  force,  passing  through  the  center 
of  gravity  of  the  recoiling  parts  and  parallel  to  the 


228 


bore,  becomes, 

i         dv 
-  [Fv-Wr(sin  0  +n  cos  0)-Rs+p-Bx)=  -rar  v  — -    (Ibs) 


GRAPHICAL  CONSTRUCTION  AND     Very  often  it  is  more 
EVALUATION  OF  THE  REACTIONS  convenient  to  evaluate 
IN  A  GUN  CARRIAGE.  the  various  reactions 

by  graphical  methods. 
Graphical  constructions 

are  of  special  value  since  they  give  a  vivid  picture 
of  the  relative  magnitude  of  the  various  reactions. 
Further  the  method  is  comparatively  simple  and  the 
closing  of  force  and  space  polygons  combined  with 
overall  methods  gives  an  admirable  check.   The  ac- 
curacy of  the  method  even  -with  rough  layouts  is  suf- 
ficient for  the  computation  of  the  various  reactions 
required. 

If  we  consider  the  kinetic  equilibrium  of  any 
piece  of  the  carriage,  we  have,  by  introducing  the  . 
kinetic  reactions  or  inertia  forces  with  the  actual 
reactions  exerted  on  the  piece,  a  dynamic  problem  re- 
duced to  a  problem  of  statics. 

For  equilibrium  of  the  piece,  we  have, 

2X  =  0   ) 

ZY  =  0    (   for  a  coplanor  set  of  forces. 
ZM  =  0    ) 

Now  the  considerations  ZX  =  0,  ZY  =  0  are  met 
by  the  vector  diagram  of  reactions  having  a  zero  re- 
sultant, that  is  the  vector  polygon  of  the  piece 
closing. 

The  condition  ZM  =  0,  requires  a  consideration 
of  the  lines  of  action  of  the  forces  in  a  space 
diagram  in  addition.   Since  the  moments  may  be  taken 
about  any  point,  there  can  be  no  resultant  moment  exist- 
ing.  The  condition  2X  =  0,  ZY  =  0  implies  the  result- 


229 


ant  force  to  be  zero,  but  does  not  imply  the 
existence  of  a  couple.   Condition  ZM  3  0. 

indies  that  a  resultant  couple  cannot  exist. 

A  graphical  method,  therefore,  always  consists  of 
two  sets  of  diagrams, 

(1)  a  space  diagram  of  forces  and 

(2)  a  vector  diagram  of  forces. 

The  space  diagram  requires  a  layout  proportional  to  the 
actual  piece  under  consideration  and  the  placing  on 
this  diagram  the  lines  of  action  of  the  forces.  The 
force  diagram  requires  a  layout  proportional  to  the 
direction  and  magnitude  of  the  various  reactions 
exerted  on  the  piece.   The  two  diagrams  must  be 
carried  on  simultaneously  since  the  direction  of  a 
resultant  required  in  a  sp^ace  diagram,  is  obtained  by 
the  vector  addition  of  the  forces  which  are  the  com- 
ponents of  the  resultant.   Since  Vector  addition  is 
commutative,  the  order  of  Vector  addition  is  immaterial. 

REACTIONS  ON  THE  RBCOILIMG  PARTS 

The  known  reactions  consist: 

(1)  The  powder  force  along  the  axis  of 
the  bore  Pfc  .     (IJbs) 

(2)  The  inertia  force  along  an  axis  parallel 

to  the  bore  and  through  the  center  of 

d  • 
gravity  of  the  recoiling  parts  -  -  mr   x's 

Zt  r   ^ 

(3)  The  weight  of  the  recoiling  parts 
acting  vertically  through  the  -center  of 

gravity  of  the  recoiling  parts  -  -  Wr. 
The  unknown  reactions  consist: 

A   a;      **   ^  **  •*•   u  £  "    * 

(1)  The   resultant  braking    force  B 

Ibs. 

(2)  The  front  and  rear  clip  reactions 
Q,  and  Q2  Ibs. 


230 


The  lines  of  -actions  of  these  forces  however 
are  known  or  can  be  readily  determined. 

Procedure: 


Layout  a  space  drawing  proportional  to  the 
dimensions  of  the  recoiling  parts,  showing  the 
assumed  lines  of  actions  of  the  various  forces. 

See  fig. (19;.       2 

Since  P^  -  »r  — —  =  K  the  total  resistance 

to  recoil  which  is 
assumed  as  known, 
we  have  the  effect  of  PK  and  m  ^  *   equivalent  to, 


b  G    "r 


dt2 


(1)  a  couple  Pfc  % 

(2)  a  force  K  through  the  center  of 
gravity  of  the  recoiling  parts  parallel 
to  the  axis  of  the  bore. 

Since  a  couple  and  a  single  force  may  always  be 
reduced  to  an  equivalent  single  force,  we  have  (1) 
and  (2)  combined  into  a  single  force  K  acting  at  a 
distance  above  the  axis  through  the  center  of 
gravity  of  the  recoiling  parts  equal  to 

P.  eh 

p  9   (in)  (  where  CK  is  in  inches) 
K 

The  reactions  0  and  Q  due  to  the  friction  in 

12  i 

the  cradle  sleeve  make  an  angle  u  =  tan  -*f  with  the 

normal  to  the  guides,  where  f  =  coefficient  of  friction 

Q 

=  0.15  usually.    Hence  u  =  8.5   approximately. 
Referring  now  to  the  force  polygon  or  diagram, lay 
off  K  in  the  direction  and  equal  to  the  magnitude  of 
the  total  resistance  to  recoil. 

Lay  off  K  =  a  b 

From  b  lay  off  b  c  =  WR,  the  weight  of  the  re- 
coiling, in  magnitude  and  direction. 

Draw  K   +  Hf  =  a  c 


231 


232 


Referring  now  .to  tha  space  diagram  lay  off  K  at  a 
perpendicular  distance  °b9b 

K 

above  the  center  of  gravity  of  the  recoiling  parts 
and  parallel  to  the  axis  of  the  bore.   At  the  intersect- 
ion of  K  and  Wp  ,  draw  J  k  parallel  to  a  c  until  it 
intersects  the  line  of  action  of  the  motion  of  the 
reaction  Q2  at  k. 

From  c  of  the  force  polygon,  lay  off  c  d  and  fn 
the  direction  of  the  rear  clip  reaction  QZ. 

Draw  k  c  from  k  to  the  intersection  of  Qt  and  8 
in  the  space  diagram. 

Draw  a  d  parallel  to  k  c  in  the  force  diagram 
until  it  intersects  a  d  at  d.  This  limits  and  de- 
termines the  magnitude  of  0  in  the  force  diagram. 

From  d,  draw  d  e  parallel  to  B  and  a  e  parallel 
to  Q  .  The  intersection  of  a  e  and  d  e  determines  B 
and  Qt  respectively.  Thus  from  a  combination  of  the 
space  and  force  diagram  we  obtain  Q2  B  and  Qt  respective- 

iy- 

REACTIONS  ON  THE  CRADLB. 


Referring  to  figure  (20): 
The  known  reactions  consist:- 

(1)  The  rear  clip  reaction  Q2     (Ibs) 

(2)  The  front  clip  reaction  Qt    (Ibs) 

(3)  The  weight  of  the  cradle  Wc   (Ibs) 

(4)  The  braking  force  B          (Ibs) 
The  unknown  reactions  consist:- 

(1)  The  trunnion  reaction  T       (Ibs) 

(2)  The  elevating  gear  reaction  E  (Ibs) 
The  direction  of  the  latter  being'  known. 

Referring  now  to  the  force  diagram  lay  off  a  b  = 
in  the  direction  and  proportional  to  the  magnitude  of 
flf.  From  fa  draw  c  parallel  and  equal  to  B  the  brak- 
ing force. 

Draw  a  c. 

Referring  now  to  the  space  diagram  J  k  from  the 


233 


in  the  force  polygon,  to  the  intersection  of  Q±  • 

In  the  force  diagram,  draw  c  d  =  Q  and 
parallel  to  Q.t  .  draw  ad. 

In  the  space  diagram  draw  J  L  parallel  to  a  d 
to  the  intersection  of  Wc. 

In  the  force  polygon  draw  lfc  equal  and  parallel 
to  Wc  the  weight  of  the  cradle.  Draw  a  c. 

In  the  space  diagram  1  m  parallel  to  a  e  to  the 
intersection  with  E  at  m. 

From  m  draw  m  n  to  the  trunnion  axis,  which  gives 
the  line  of  action  of  the  trunnion  reaction  T. 

In  the  force  polygon  draw  e  f  in  the  direction 
of  E  and  a  F  in  the  direction  of  T.   The  intersection 
at  f  determines  the  magnitude  of  E  the  elevating  gear 
reaction  and  T  tha  trunnion  reaction. 

t»  yf  "X 

REACTIONS  OH  THE  TIPPIHG  PASTS. 


Locate  the  trunnions  along  the  resultant  of  the 
"battery  position  of  Wr  and  Wc  ---  See  upper  diagram. 

Without  balancing  gear:- 

Considering  the  external  forces  on 
tipping  parts,  we  have,  the  known  reactions, 

(1)  The  total  resistance  to  recoil  K  (l"bs) 

(2)  The  weight  of  the  tipping  parts  Wt 
(Its) 

The  unknown  reactions, 

(1)  The  elevating  gear  reaction  E  (l"bs) 

(2)  The  trunnion  reaction  T      (l"bs) 
the  direction  of  E  "being  known. 

In  the  space  diagram  lay  off  X  parallel  to  the 
bore  and  at  a  perpendicular  distance  from  the  center 
of  gravity  of  the  recoiling  farts  =  pbe   (.  . 

In  the  force  diagram,  lay  off  ab  =  K  and  be  =  Wt. 

Draw  ac. 

In  the  space  diagram,  lay  off  J  k  from  the  inter- 
section of  K  and  Wt  parallel  to  ac  of  the  force  diagram 


234 


235 


I 


<L 
6 

U 

o 

o 
t- 


0 -o 


& 

5 

\ 


u 
o 

i 

of 

ft: 
a: 
o 

0. 

o 

K 


cu 

V) 


236 


a: 
Q 


O 


d 

I 

0 

U 


237 


£ 
<c 

s 

cr 


uJ 
u 
ot 
o 
u. 


I 


Q 

LJ 

o 

d 

0. 
CO 


u           a 

(3 

m 

£ 

K 

ID 
N 

<t 

DO 

0 

iZ 

Q 

*^'pfe;   k  ' 

y 

3 

^*                    ^ 

-    nl/f     r  x 

b 

i 

/    7  '{           8^ 

\ 

£ 

/  A         §5 

\ 
\ 

i 

0 

c 

/  \/\  i       g 

/y  '  t 

/7     !       s 

\ 
\ 

\ 
\ 
\ 
\ 

f   / 

\ 

x/    / 

\ 

I 

/  /    / 

i 

'  /   / 

*/  ' 

'// 

^-X'/ 

/   /       " 

/  //  /7  / 

Aw  !' 

Its 

nr^ 

/  A 

^  >     \  1  Z              o 

^^-"' 

I  r>- 

238 


and  t o  the  intersection  of  E. 

Draw  k  1  to  the  trunnion  axis  in  the  space  diagram. 
The  line  of  action  of  T  is  then  along  h  1  produced. 
In  the  force  polygon,  draw  cd  parallel  to  E  and  a  d 
parallel  to  k  1  of  the  space  diagram.   Their  inter- 
section at  d  determines  the  magnitude  of  £  and  T 
respectively. 

With  balancing  gear:- 

Determi nation  of  the  balancing  gear  reaction 
R.   On  the  space  diagram  lay  off  Wj.  the  weight  of  the 
tipping  parts  in  its  battery  position  as  well  as  the 
line  of  section  of  R.   From  the  intersection  of  R  and 
Wt  draw  o  m.   This  must  be  the  direction  of  the  result- 
ant of  Wt  and  F  since  the  condition  is  that  we  have 
no  moment  about  the  trunnions  when  W^  is  in  its  battery 
position. 

In  the  diagram  below,  lay  off  W.  and  R  and  draw  o.m 
parallel  to  o  m  in  the  space  diagram.   This  determines 
the  magnitude  of  the  balancing  gear  reaction  R. 

Referring  to  the  force  diagram,  lay  off  a  b  equal 
and  parallel  to  K  the  total  resistance  to  recoil,  and 
be  =  Wt  the  weight  of  the  tipping  parts. 

Draw  ac. 

In  the  space  diagram  K  is  at  a  perpendicular  dis- 

D   p 

tance  _fc_-  from  the  center  of  gravity  of  the  recoil- 
ing  parts  and  V^  at  a  distance 

Wrx  cos  0 
— * from  its  battery  position,  where 


x  is  the  displacement  in  the  recoil 

At  the  intersection  of  K  and  Wt  draw  j  k  parallel  to  ac 
to  the  intersection  of  R. 

In  the  force  polygon  draw  cd  parallel  and  equal  to 
F.   Draw  a  d. 

In  the  space  diagram  draw  1  k  parallel  to  a  d  of 

the  force  polygon  to  the  intersection  of  the  line  of 
action  of  E  ac  1.   Draw  1  m  to  the  trunnion  axis,  thus 
determining  the  line  of  action  of  the  trunnion  re- 
action T. 

In  the  force  polygon  draw  d  e  parallel  to  m  E  and 


239 


a  e  parallel  to  1  m,  thus  determining  the  magnitude 
of  E  and  T  respectively, 

R3ACTION3  ON  THB  TOP  CARBIAGB 


Without  balancing  gear:- 

The  known  reactions  consist: 

(1)  The  weight  of  the  top  carriage 
Htc  (Its) 

(2)  The  trunnion  reaction  T   (Ibs) 

(3)  The  elevating  gear  reaction  E 
(Ibs) 

The  unknown  reactions  consist: 

(1)  The  horizontal  component  of  the 
pintle  reaction  -  H   (Ibs) 

(2)  The  vertical  component  of  the 
pintle  reaction  N   (Ibs) 

(3)  The  front  vertical  clip  reaction  M 
(Ibs) 

The  lines  of  actions  of  these  forces  are  given 
from  the  construction  of  the  piece. 

Referring  to  the  force  polygon  fig. (25),  draw  ab 
=T  equal  to  the  magnitude  and  in  the  direction  of  T 
the  trunnion  reaction.   Draw  be  parallel  and  equal  to 
E  the  elevating  gear  reaction. 

Draw  ac.   In  the  space  diagram  draw  j  k  parallel 
to  T. 

At  the  intersection  of  j  k  and  E  produced  draw  k  1 
parallel  to  a  c  in  the  space  diagram  to  the  intersection 
of  Wtc. 

In  the  force  polygon  draw  c  d  equal  and  parallel 
to  Wtc-   Draw  a  d   Prom  1  in  the  space  diagram  1  IE 
parallel  to  a  d  to  the  intersection  of  N  produced. 

From   m  draw  m  n  to  the  intersection  of  H  M.  Draw 
a  e  in  the  force  polygon  parallel  to  mn  in  the  space 
diagram. 

We  thns  have  d  e  in  the  force  polygon  =  N  and 
ef  =  M  and  ja=  H. 


240 


Thus  the  pintle  reactions  H  and  N  and  the  clip 
reaction  are  determined  in  magnitude  and  direction. 

With  balancing  gear:- 

The  "known  reactions  consist: 

(1)  The  weight  of  the  top  carriage  Wtc 

(2)  The  trunnion  reaction  T 

(3)  The  elevating  gear  reaction  E 

(4)  The  balancing  gear  reaction  B 
The  unknown  reactions  consist: 

(1)  The  horizontal  component  of  the 
pintle  reaction  H 

(2)  The  vertical  component  of  the  pintle 
reaction  N 

(3)  The  front  vertical  clip  reaction  M 
The  lines  of  actions  of  these  forces  are  given 

from  the  construction. 

Referring  now  to  the  force  polygon  fig. (24)  Lay 
off  ab  =  T  and  be  =  E.  Draw  ac. 

In  the  space  diagram  from  the  intersection  of 
T  the  trunnion  reaction  and  E  elevating  reaction  pro- 
duced at  K. 

Draw  k  1  parallel  to  ac  of  the  force  polygon.   Con- 
tinue in  the  force  polygon  c  d  =  R  the  balancing  gear 
reaction.   Draw  ad.   In  the  space  diagram  draw  in  parallel 
to  ad  and  the  intersect!  on  of  W^  at  m.   In  the  force 
polygon  draw  de. 

Draw  ae . 

In  the  space  diagram  draw  mn  parallel  to  ae  to  the 
intersection  of  N.  Fron  N  draw  n  o  to  the  intersection 
of  M  and  H.   In  the  force  polygon  draw  a  f  parallel 
to  o  n.   From  E  in  the  force  polygon  draw  e  f  parallel 
to  N  to  the  intersection  of  e  f . 

Draw  f  g  and  &  a  as  shown. 

Thus  we  determine  the  reactions  M,  N,  and  H 
respectively. 


241 


REACTIONS  ON  THE  ASSEMBLED  CARRIAGE  GUN  ASP  CARRIAGE 
TOGETHER. 

Location  of  the  weight  of  the  total  mount:- 

Assuming  a  static  reaction  of  200  Ibs.  under 
the  spade,  we  lay  off  o'm  =  200  Ibs. 

Then  o  N  =  Wg  =  200  under  the  wheel  contact. 
The  resultant  of  o'm  and  o  n  »  W3  obtained  by  the 
additional  construction  lines  o'q  and  op.   Hence  we 
determine  from  the  triangle  of  forces  the  line  of 
action  of  *L.   The  external  reactions  on  the  as- 

9 

sembled  carriage  consists  of  :- 
The  known  reactions  - 

(1)  K  =  the  total  resistance  to  re- 

coil. 

(2)  Ws  =  the  weight  of  the  total  mount. 

The  unknown  reactions  - 

(1)  The  horizontal  spade  reaction  Ha. 

(2)  The  vertical  spade  reaction  Va. 

(3)  The  normal  reaction  under  the  wheels 


The  direction  of  these  forces  are  known. 

Referring  to  the  force  polygon  lay  off  ab  =  K  the 
total  resistance  to  racoii  and  be  =  weight  of  total 
system  W3. 

Draw  ac. 

In  the  space  polygon  from  the  intersection  of 

<  and  Wg  draw  j  k  to  the  intersection  of  the  reaction 

Va- 

Prom  k  draw  k  1  to  the  intersection  of  Ha,  Vb 

at  1. 

Referring  to  the  force  diagram  draw  ad   parallel 
to  1  k  of  the  space  diagram  to  t"hs  intersection  of 
c  e  produced. 

We  thus  determine  c  d  -  Va,  d  e  =  Vb  and  e  a  -  Ha< 


242 


Thus  the  reactions  Ha,  Ba  and  V^  are  determined 
in  magnitude. 

PROCEDURE  IN  THE  CALCULATIONS  FOR  THE  PRINCIPLE  RE- 
A.CTIONS  IN  A  GUN  CARRIAGE  MOUNT. 


(Illustrated  by  calculations  on  240  n/m  Hewitmer) 
REOUIRBD  DATA. 

Type  of  Gun  Howitzer 

Diameter  of  bore  d  (in)  9.45 
Total  Weight  of  recoiling  parts  Wr(lba)    15790 

Weight  of  Powder  Charge  W  (Ibs)  40 

Muzzle  Velocity  v  (ft/sec,)  1700 

Travel  of  Shot  in  Bore   u  (in)  160 


maximum  60° 

Angle  of  Elevation  0  ninimum  10° 

short  3.74 

Length  of  Recoil   b  (ft)   long  3>g0 


Intensity  of  Powder  Pressure  p^dbs/sq.in)  32000 

Initial  Air  Volume  of  Recuperator  Vai       2970 

(cu.in) 

Initial  Air  Pressure  of  Recuperator  Pai      576 

(Ibs/sq . in ) 


243 


INTKRTOR  BALLISTICS. 

Maximum  Powder  Pressure  on  Breech         2,245,000 
F  »  Pb  =  0.7854  d2pm  (Ibs) 

Maximum  Powder  Pressure  on  Base  of        2,005,000 
Projectile  pm  (Ibs) 

P°=A:   (Us) 


Mean  Constant  Powder  Pressure  jQ  *  *7°Q 

5.36  x  160 


5*36U  1,350,000 


1  =  twice  abscissa  of  Max.  Pressure 

-  ~  —)*  -  1       3.996 


POD  =  Muzzle  Pressure  on  base  of  breech 


622,000  Ibs. 


Vsl.  of  free  recoil:  7f 

wVm  +  4700  W 

=  50.25  ft/sec 

Wr 


Vel.  of  free  recoil  -  Shot  leaving 
Muzzle 


0.5W  Vm 


wr 


40.50  ft/sec, 


Time  of  Shot  to  Muzzle 

t  s  —  — i-  0.01175  sec. 

1   2  12V,, 


244 


Time  of  Expansion  of  Free  Gases 
- 


ob 


32.2 


0.01538  sec, 


Free  Movement  of  Gun  while  shot 
travels  to  Muzzle 

•_  u"(w+0.5W) 

l~  12(Wr+w  +  w) 


0.31  ft. 


Free  Movement  of  gun  during  Pow 
der  Expansion 
P  v    t* 


0.7179  ft. 


Total  free  Movement  of  gun;  Pow- 
der Pressure  Period: 
I  •  Z4  +  X, 


1.0279  ft. 


Time  of  Powder  Pressure  Period 
r  -  tt  *  tf 


0.02713  sec, 


BRAKING  PULLS  AMD  STRESSES  IN  CYLINDERS. 


x  axis  taken  along  bore:  v  axis  taken  normal  to  bore. 


Mass  of  Recoiling  parts 

»r 
"r  ''  32.16 


15790 
32.16 


491 


Constant  of  Stability 
C  »  0.85  to  0.9 


Calculations  only 
for  max.  elov. 


Height  of  center  of  gravity  of 
recoiling  parts  above  ground 
h  (ft) 


Calculations  only 
for  max.  elev. 


Stability  Slope 


elf. 


Calculations  only 
for  max.  elev. 


245 


Total  Resistance  to 

Max.Elev. 
Recoil  


491  *  50.75 


Hor.Elev. 

,2 


K  = 


2(b-£+VfT) 


(Ibs) 


2 (3. 74-1. 0279+50. 75*. 02713) 
152,000 


Variable  Resistance  to  re-  Calculations  only  for  max. 
coil  in  battery  (at         elev. 
horizontal  elev.) 


jnV 


K-- 


rf 


2[b-E+VfT-  -  - 
2  M, 

(Ibs) 


Variable  Resistance  to  Re-  Calculations  only  for  max. 
coil  out  of  battery  (at     elev. 
horizontal  elev.  ) 


k  =  K-m(b-E+  -  ) 
2m., 


Initial  Recuperator  Re- 
action,  Pai  =  approx. 
1.3Wr(sin  0m+0.15cos0m) 
Ibs.  (unless  given) 


1.3  x  15790  (gin  60+0. 15cos60) 
=  19300  used 
18800  Ibs. 


Total  Initial  Recuper- 
ator Pull,   Pai  =  P^i 


100  d. 


(Ibs) 


18800+100x2.938 


19094 


da  =  diam.  of  recuperator 

rod. (in) 

0. 

Effective  Area  of  Recuper-  35.756 
ator  Piston  - 
Aa  (sq.in) 


Initial  Air  Pressure 


ai 


(Ibs/sq.in) 


18800 
32.6 


576 


246 


Initial  Air  Volume  Vai  (cu.in)    2970 


Final  Air  Volume  Vaf(cu.in) 
vaf  -  Vai  -12  Ab 


a 


Final  Air  Pressure 
P.,  "".'" 


Final  Recuperator  reaction 


2970  -  12  x  32.6  x 
3.74  =  1510 

S76 

1214 

1214  x  32.6  =  39600 


af 


=  p 


af  a 


air 


Paf=  approx.  2Pai(lbs)metallic 

J_Distance  from  axis  of  bore  to  3.038+  3.850 
mean  guide  contact  r(in) 


3 
3.4444 


Distance  between  clips  1  (in)    86.25 

J.  Distance  from  axis  of  bore  to  16.365 
center  line  of  hydraulic  pis- 
ton e^  (in) 

J_Distance  from  axis  of  bore  to 

line  of  action,  of  recuperator    15.656 

reaction  ea(in) 


Assumed  coefficient  of  guide 
friction  u  =  0.15  to  0.25 

Guide  friction  constant 

2u 

»  Af 

l-2ur 


0.15 


0.15 


86.25  -2x0.15x3.44 
.00352 


247 


Total  hydraulic  Pull      152000+157908in60-18800(  1+.0635) 
(max.  elcv.)  1+.0663 

»  137500 


UAfeh      (Ibs) 

Total  hydraulic  Pres-     2  hydraulic  cylinders: 
^-100  dn   ^  ' 

diam.  of  brake  rod 


Ph»P^-100  dn   ^          68750-4.72x100=  68280 


(in) 

Effective  Area  of  Hy-     31.2 
draulic  Piston 


Max.  Pressure  in  Hy-     68280 

-  =  2200 
draulic  Cylindsr          31.2 

'      (Ibs/sq.in; 


Inside   Diam.    of  brake          7.874 
cylinder 


dih-  1, 

(in) 

(dn*  diam.  recuperator 
rod) 

Outside  diam.  of  brake   9.450 
cylinder 


Hoop  tension  in  brake 

cylinder  wall          2200(==  —      )  =  12150 

9-45'  -7.875' 


Ibs/sq.in. 


248 


Max.    pressure   in  recuperator        1214 
cylinder       v 


Paf=?ai 


Ubs/sq.in) 


Inside    Diam.    of   recuperator          7.087 

cylinder 

dia   =    1.13/A0+0.785d|    (in) 

Outside    Diara.    of    recuperator        8.267 

cylinder 

doa     (in) 


Hoop  Tension   in  Recuperator 
Cylinder  Wall 


,d§a+d!a 

1  d 2   -d  *    ' 
uoa  uia 


.  in) 


.8.267+7.087 
12140==1)  =8020 


.267-7.087 


Inside  Diam.  of  compressed 
air  storage  tank  d   (in) 


84-66 


Outside  diam.  of  compressed 
Air  Storage  tank  doc  ^in^ 

Hoop  Tension  in  compressed 
air  storage  tank 


Paf  ( 


9.45 


a  —  —  __2 

1214  C—^—  *  '   —  ) 
9.45-8.466' 


11000 


d2    -d2. 
oc      ic 


(ibs/sq.in) 


Width   of   Wall    between  ad- 
jacent   cylinders^   (in) 


Hoop   tension  between  adjacent   -  -  -  -  - 
cylinders 

p   =   phdih*Pafdia 


1.8  w 


.i  n) 


249 


GUIDE,  ELEVATING  GEAR  AND  TRUNNION  REACTIONS: 

'. 

x   axis    taken    along  bore:  v    axis   taken  normal    to 

bore. 

Coordinates  from  center  of      Xs  37.843 
gravity  of  recoiling  parts 
to  front  guide  reaction         yt*-3.038 
xt  and  yt  (in) 

Coordinates  from  center  of      xg  »  48,4O7 
gravity  of  recoiling  parts  to 
rear  guide  reaction  ya=  3.86 

x,  and  y2(in) 

J_  distance  from  center  of      16,365 
gravity  of  recoiling  parts  to 
brake  piston  rod  axis  e^ 


J_  distance  from  center  of      15,656 
gravity  of  recoiling  parts  to 
recuperator  piston  rod  axis 
ea  (in) 

•U^esa-r -:-™*       U    (  $*  o 

Max.  powder  reaction  P^T  (Its)   2,245,000 
(See  Interior  Ballistics) 

J_  distance  from  center  of       6.13 
gravity  of  recoiling  parts  to 
axis  of  bore  e  (in) 


Front  guide  reaction:  gun  re- 
coiling in  sleeve: 

Fe+Pneh+Pa-Wrcos0(x2-uy2) 

Q   5S  I  - 


0.15  to  0.2    (Ibs) 


250 


Rear  guide  reaction:  gun  recoil- 
ing in  sleeve 


Q  =• 


Fe+P£en+Paea+Wrcos 


(Ibs) 
Front  guide  reaction:  gun  recoil-  2,245,000x5.13+137500 


ing  in  guide  below  axis  of  bore    37. 84+48. 41-0. 15><6. 91 
Fe+P£en+Paea-Wrcos0(x8-uy8) 


t  +  x  -u(yt+y 


Rear  guide  reaction:  gun  recoil- 
ing in  guides  below  axis  of  bore 


Q 


xl6.  365+19094x15.  66 


-7895x47.63 


2,245,000x5.13+137500 
37.84+48.41-0-15x6.91 


x!6.  365+19094x15.  66 


+7895x37.38 


162600 


Max.  guide  friction 
Kg  »  u(0t+a8)-  (Ibs) 
u  =  0.15  (approx. ) 

Weight  of  Tipping  Parts  Wt(lbs) 

Max.  Resistance  to  recoil  (dur- 
ing powder  period) 


2Peu 


Bg  =  0.15(154800+ 
162600)=  47,620 


21,021 

=137500  +  19094+47,  620- 
13670=191000 


=152000+ 
5.13x0.15 


2x2,245,OQQx 
85.21 


=192000 


251 


I  distance  from  trunnion  axis   3.73 
to  line  parallel  to  axis  of 
bore  through  center  of  gravity 
of  recoiling  parts  s  (in) 

Radius  to  pitch  circle  of        35.57 
elevating  arc.  j  (in) 

Angle  between  "y"  axis  and  the   60° 
radius  to  elevating  pinion  con- 
tact with  elevating  arc  9e  * 
0  +  ne 


Elevating  gear  reaction  (in 
battery)  E  -  Fe]K'a.  (Ibs) 
Angle  of  E  with  horizontal 


Top  carriage  trunnion  reaction 
(in  battery  with  balancing 
gear) 
2X=K+Wr3in0+Bcos99+Rsin9r  (Ibs) 

2Y=Htcos  0+Esinee-Rcos9r   (Ibs) 
(E  is  sans  with  or  without 
balancing  gear) 


Top  Carriage  Trunnion  reaction   Not  used. 

(out  of  battery  with  balancing 

gear) 

2X=K+Rsin9r+B'cos9e+Wtsin  0  (Ibs) 

2Y=Wtcos0+E'sin9e-Rcos9ii    (Ibs) 

(E1  is  same  with  or  without 

balancing  gear) 


11,513,884*191,000x3.73 
35.57 

»  344,000 
Not  used. 


Estimated  Weight  of  Bocker 
W-  (Ibs) 


Neglected  as  small 


252 


Horizontal  Distance  from  Trunnion 
to  center  of  gravity  of  rocker 

hr  (in) 

"measured  to  rear" 


Not    used. 


Angle   between    line   of    action   of 
rocker   reaction   on   cradle   and 
"y"    axis.      B 

I     distance    from    trunnion    to 
elevating    sere*   or    normal    to 
rocker  cradle  contact, 
k   =   x^os  B+y^sin  B         (in) 
"xm    and  ym  coordinates    of 
rocker  contact  with   cradle 
from   trunnion   to   rear   and 
down" . 


+  30 


29. 43x. 866+15. 71 
*0.5  -  33.35 


Rocker  Reaction  on  Cradle 
M=  -^— —     (Ibs) 


344000^35.57 
33 . 35 

367000 


Elevating  Bear  Reaction 
(out  of  battery) 


E1  = 


Ks+Wrbcos  0 


(Ibs) 


Calculations  max, 
elev.  in  battery, 


Top  carriage  trunnion  reaction 
(in  batterv)(X  and  Y  components) 
2X=K'+lft3in  0+E  cos  6e   (Ibs) 
0-E  sin  6 


2X=197000+18200+ 
17200=381200 
2Y=10510-198,000= 
-287,500 


253 


Top  carriage  Trunnion  Reaction 
(out  of  battery)  (X  and  Y  com- 
ponents ) 

2X=K+Wr   sin  0+   E  cos   9e    (Ibs) 


cos  0-Ein 


(Ibs) 


Calculation  at  max. 
elevation   in  battery, 


Vith  balancing  Gear:      Distance 
from  trunnion  to   center  of 
gravity  of   tipping  parts    (in 
battery)   along  x   axis: 
xt      (in) 


Not   -used. 


Radius   of  bell   crank 
(balancing   gear) 
r,,      (in) 


Not  used. 


Balancing  Gear  Reaction: 
2Wtxt  cos  0 


ra(l+cos 


(Ibs) 


Not  used. 


(very  approx.) 
or  calculated  from  layout 
0m  =  max.  elevation. 


Angle  made  by  balancing 
gear: reaction  with  "y" 
axis 


9, 


(See  layout) 


Not  used. 


Rocker  Trunnion  reaction 

(X  and  Y  components) 

2Xr  =M  sin  B-E  cos  8e~V* 

sin  0     (Ibs) 

2Yr=  E  sin  9g  -¥*  cos  B 

(Ibs) 


183500  -  172000 
11500  =  2Xr 
297000  -  318000 
-  21000  =  2Yr 


254 


Total  shear  reaction  of  trunnion   190600  +  5750 
on  cradle,  -  X'=X+Xr   (Ibs)        196350  =  X1 

Y'=Y+Yr   (Ibs)        -143750-10500' 

154250  =  Y1 


Total    spring   reaction   of  Top 
Carriage   on    trunnion 


sin 


(Ibs) 


cos  0   (Ibs) 


10000  *  .866 
8660  =  X3 
10000  *  o.S  = 
5000  -  Ye 


Total  rigid  bearing  reaction  of 
top  carriage  trunnion 
Xb  =  X  -Xs   (Ibs) 
Yb  =  Y  -Ys  (Ibs) 


190600  -  8660 
181940  =  Xfc 
-143750-5000= 
-149750=  Y, 


Bending  moment  at  cradle  section   8660  x  5.5  + 
of  trunnion  181940  *  2.9  +  5750 

Mx=  Xsa  +  Xbb  +  Xrc  (Ibs)         *.0.9  ~  580780 
My  =  Ysa  +  Ybb  +  Yrc  (Ibs;        5000  x  5.5  -  149750 

*2.9  -  10500  x  0.9  = 
-416,950 

Resultant   B.   W.    at   cradle   section 
of   trunnion 


(in   Ibs)  /580,7802    +   416,960' 

716,000 


Max.    fibre   stress   due    to  bend- 
ing 

a    10.18   M  (Ibs/sq.in) 


10.18   x    716000 
—8* 


355 


n 


rJ 
evi 


7.5 


3.6  — 


TfcVJNN\QN     PIN 


~  n  -. 


256 


SHEAR   REACTION  OF  CRADLE. 
ON  TRUMNtON  PW  : 


RERCT\ON  OF  ROCKER  ON 
TRUNNION  P\N : 


RERCT\ON  OF  TOP  CRRR\RG£. 
ON    TRUNNION 


\9O6OO 
RERCT\ON  ON    P\N    \N    X    PURNE.. 


257 


258 


CALCULATIONS  FOR  STRENGTH  OP  CARRIAQ1  AXLE 


Proposed  75  m/n  St.Chamond 


50°  Elevation   and 


22-   traverse: 


Maximum  Peak  Resistance  to  Recoil  -  -  assumed 

at  20,000  Ibs. 

The  resistance  to  recoil  may  then  be  divided  into 
a  horizontal  and  vertical  component  in  the  vertical 
plane  of  traverse.   T"hen,  the  horizontal  component  in 
the  vertical  traversed  plane,  nay  "be  divided  into  a 
component  along  the  horizontal  axis  of  the  mount  and  a 
transverse  component  at  right  angles  to  the  longitudinal 
axis  of  the  mount. 

The  components  in  the  vertical  traversed  plane 
are:- 

Horizontal  comp.  =  20,000  *  cos  50°  =  12820  Ibs. 
Vertical  comp.  =  20,000  *  siii  50*  =  15320  Ibs. 

The  longitudinal  and  transverse  "horizontal  com- 
ponents are:- 

Horizontal  comp.  =  12820  *  cos  22.5°  =  11800  Its 
Transverse  comp.  =  12820  *  sin  22.5°  =  4900  Its. 


859 


Then,  15320  +  4000  =  19320  (Total  Downward  Force) 

S  x  130  =  4000  x  120.25  +  15320  *  128.2  -  11820  *  47.2 
4000  x  120.25  =  481000 
15320  x  128.2  =  1970000 


11820  x  47.2  = 

19320 

14550 

4770 


2451000 


558000 
1893000 


S  =  14,550 


4,770 


A,  +  B2  =  4770 
»nef 


260 


12800   x  cos  *2     =   11800 


12800  x  Sin22-  *   4900 

2MA  »  8X  x  142.4  -11800  x  71.2  +  4900  x  128.2  »  0 

11800  x  71.2  =  840000 

4900  x  128.2-  629000 

211000 

H800  ...  B  •  2110°°  •  1481   ) 

1481  L42-4         ( 

A,  =  10319 


£M  about   vertical   pin   for   loft   trail 
Ay   126.38  *   10319    x  55.2 


4900  .'.      A,    4500 

4500 

400  By   -      400 


IM  axle   -   A,   130  -10319    x  32  *   Ba   130  -   1481 
130C/1,  -8Z)  =  10319   x  32  -  1481   x  32 
=283000 


.'.  AZ-BZ  =  2180 


Bz=   4770 


2AZ        -      6950  .*.  Az   =   3475          ) 

8Z   =    1296          j 

£M  about  left  wheel  base  in  Z  Y  plane: 

-4900  x  41.2  +  15300  x  30  +  4000  x  30  -  4900  x  6 

+  3475  x  41.2  -  12.95  x  101.2  -  Sp  x  60  »  0 

-  4900  «  41.2  *  -  202000 

49000  x  6   »  -  29400 

-1295  x  101.2  =  -  131100 

-  362500 


261 


15300  x  30  *  459000  722000 

4000  x  30  =  120000  362500 

3475  x  41.2=143000  359500 
722000 

Sg  =   5980   ) 

SA  =   8570   ) 

Reactions   on  Trail   Axle. 

X  and   Y  reaction  on  vertical   pin  of   left   trail: 

Ex  =   10319  t  Ey  =   4500 

B.   ¥.    in  XY  plane   on  axle: 

Ey   x   10  =   4500   x   10  =   45000    "   #       XY  plane: 

Thrust   along  X  axis   =   10319  ±   Shear  reaction  of  equal- 
izing bar. 

Thrust   along    Y  axis   =   4500 

Thrust   along   Z  axis   =  3475 

Shear   reaction  of  Equalizer  bar  = 
452000  -  331000 


7.75 
Thrust  along  K  axis 


15600 

10319 
15600 

25919 


EXTERNAL    FORCE.S   ON   RXUE 
FOR    SECTION  -(m-n 

& 


-45OOO 


n 


262 


Section  n-n  5"  x  5" 

Torsion  =  25919  x  2.2  =  57000  ("  *) 

(B.M.zy)  =  3475  «  12.2  +  8570  x  2'6 

42300 

22300Q       B.  M.  jjTy  =  265300  "  * 
265300 
(B.M.zy)  =  25919  x  12.2  -  45000 

316000 

45000          B.  M.  =  271000  ("  *) 
~27100(T 


f 


I   \  *  30032-  •=  Ci  >  :'0---.  *  01  *  ^1 
265300  x  2. 


.  12700 


5  x  25 

271000  x  6       13000 
fy  '     125         "  25700 

n  54     625  n 
32      32 


.  46500 
•1.4 


12850  +  S\  x  257002  +  46502 
12850  +  13620  =  26,470 

BICAPITDLATION      Qf      gQ  R  HUT.  AE      OH      THR      TNT^BKAT. 
BBACTIOHS     THROliaHnilT      A     GHH     C*RBT«GB. 

F   =   Powder   reaction  (Ibs) 

B   =  Total   braking    force    not    including 

guide    friction  (Ibs) 


263 


"b  =  distance  from  center  of  gravity  of  recoiling  parts 
to  line  of  action  of  8.  (in) 

R  =  total  guids  friction  (Its) 

r  =  mean  distance  from  center  of  gravity  of  recoiling 
parts  to  guide  friction  (in) 

e  -  distance  from  center  of  gravity  of  recoiling  parts 
to  line  of  "bore.  (in) 

Pn=  total  oil  pressure  on  the  "hydraulic  piston.   (Ibs) 

P'=  the  hydraulic  reaction  plus  the  joint  frictions 
^   (stuffing  box  at  pistons)  (Ibs) 

Pa=  the  total  elastic  reaction  (due  to  compressed  air 
or  springs)  (Ibs) 

Pa*  the  total  elastic  reaction  plus  the  joint  frictions 

(Ibs) 
Cj,  3  distance  from  center  of  gravity  of  recoiling 

parts  to  line  of  action  of  Pn.  (in) 

ea  =  distance  from  center  of  gravity  of  recoiling 

parts  to  line  of  action  of  Pa.  (in) 

d},  =  stuffing  box  or  rod  diam.  of  hydraulic  cylinder. 

"V5  (in) 

da  -  stuffing  box  or  rod  diam.  of  air  cylinder.   (in) 
Q  =  normal  front  guide  reaction  (Ibs) 

0  =  normal  rear  guide  reaction.  (l"bs) 
xt  and  yt  =  coordinates  from  center  of  gravity  of  re- 
coiling ]!>arts  to  front  guide  reaction,  (in) 

1  =  distance  between  line  of  action  of  Qt  and  Qf  (in) 
x   and  yf  =  coordinates  from  center  of  gravity  of  re- 
coiling parts  to  rear  guide  reaction,  (in) 

!fr  =  weight  of  recoiling  parts.  (Ibs) 

0  =  angle  elevation. 

u  =  coefficient  of  friction. 

X.  and  Y  =  component  trunnion  reactions  (Ibs) 

Xr  and  Yr  =  component  roc"ker  reactions  at  the 

trunnion  (Ibs) 

&  =  elevating  gear  reaction 

J  =  radius  to  pitch  circle  of  elevating  arc.     (in) 
9a=  angle  between  "y"  axis  and  the  radius  to  elevating 
pinion  contact  with  the  elevating  arc. 


264 


K  *  total  resistance  to  recoil.  (Ibs) 

s  »  distance  from  center  of  gravity  of  recoiling  parts 

to  trunnion  axis  measured  along  the  "y"  axis,  (in) 

Total  resistance  to  recoil  on  recoiling  vass.  becomes. 

K  *  B  +  R  -  Wr  sin  0  (Ibs) 

but  B  =  Pn  +  Pa 

where  Ph  =  Ph  +  100  dn       )  assu>ing  100  1T)8.  pCr 

and     i  i        (in  diaro.  for  frictions 

P.  »  P.+  100  d. 

)  in  stuffing  box. 

hence 

K  *  P  *  ?  +  R  -  Vi  sin  0 


QUIDK  OR  CLI?  »KACTIQ«8  TO  QUIDS  FRICTIOM. 

Gun  recoiling  in  sleeve,  front  guide  reaction, 

Fe+Bb-W_   cos  0(x.   -uy.  ) 

Qt   -  -  -  -  '  -  *—  (Ibs) 

xt+xf+u(yt-yt  ) 

and   rear  guide    reaction. 

• 

Fe-»-BbCWr  cos  0  (x±*  uy,.  )  /1V  . 

\  (Ibs) 


Gun   recoiling    in  guides  below  the    axis   of   the  bore. 
front  guide    reaction, 

Pe+Bb-W.cos  0   (x  -uy    ) 
Q     =  -  -  -  -  -  2_  (Ibs) 


and  rear  guide  reaction, 

Fe+Bb+Wr  cos  0(x  -uy  ) 

Q^   =  -  -  -  -  -  V      *  (Ibs) 

«,+    x4-u(yf+ya) 

If  R  *   xt+xt+u(yt-ya  )      for   sleeve   guides 

M  =   xt  +  xa-u(yi+y2  )     for  guide  below   axis   of  bore 
and 


265 


H  *  x  -x  +H  (yt+yt  )     for  sleeve  guides 

*  *  xt-xt+u(yt-yt)     for  guides  "below  axis  of  bore. 

then  the  total  guide  friction  equals, 

2(Fe+Bb)+W     cos  0   .    N 
R   =  -  -  -  u  ,(lbt) 

and  for  the  total  braking  force  B4 

(K+W_  sin  0)M-(2Fe+W_  cos  fS  N)  u 
B  =  -  1  -  1  -     (It.) 

X  +2  u  b 

In  terns  of  tbe  pulls,  we  bave  for  the  clip  re- 
actions, 

Fe  +  £Pa  +  2Pe~lf  cos  ^(x~u) 


Q     ,  -  —  —          (Ibs) 
*t**t*u<*t-y«> 

Fe+IPaea+    2Pv«h+WP   cos  0(x    +uy    ) 

"     -  i  -  J-    (Ibs) 


xt+ 
and  tbe  guide  friction  becomes, 

2Fe+22P^eh+  22Paea+  Wr  cos  0  K 
R  =  -     (Ibs) 
M 

and  tbe  hydraulic  pull  in  terns  of  the  total  re- 
sistance to  recoil  and  recuperator  reaction,  becomes, 

M(K-£Pa~Wr  sin  0)-u(2Fe+22P'efl+  N  *fr  cos  0)  , 

«  A      r  -  A.  a      .     .,  r_  ,  _<  (Ibs) 


For   approximate  calculations,  the  guide  friction 
equals,   2u8dr 

R  "     ~ 


From  tbe  foregoing  analysis  we  observe,  that  tbe 
guide  friction  and  bearing  pressures  are  reduced: 

•  (1)     By  increasing  ths  distance  between 
the  clips. 


266 


(2)  By  balancing  the  pulls  about  the 
center  of  gravity  of  recoiling 

parts  or  bringing  the  resultant  pull 
closer  to  the  center  of  gravity  of 
the  recoiling  parts. 

(3)  By  "bringing  the  resultant  friction 
line  of  the  guides  closer  to  the 
center  of  gravity  of  the  recoiling 
parts  . 

(4)  By  reducing  the  powder  pressure 
couple  Fe,  that  is  by  reducing  the 
distance  from  the  center  of  gravity 
of  the  recoiling  parts  to  the  center 
line  of  bore.  The  distance  from  center 
of  gravity  of  the  recoiling  mass  to 
the  center  line  of  bore  should  never 
exceed  1.5  inches  unless  a  friction 
disk  is  introduced  with  angular  notion 
about  the  trunnion. 

Stress  QB 


Let 

Wc  =  weight  of  piston  and  rod  or  the  weight  of 
recoiling  cylinder.  (Ibs) 

d-  =  distance  from  center  of  recoil  pull  to  section 

"mn"   adjacent  gun  of  the  gun  lug.   (in) 
Imn  =  moment  .of  inertia  of  section.      (in)' 

y  -  distance  to  extreme  fibre  from  -neutral 

axis.  Cin) 

fnn3  nax.  fibre  stress  (Ibs/sq.in) 

then,         W' 

[B+  -2.  (F-B)]dgy 


-n 


(Ibs/sq.in) 


Trunnion  and  Elevating  gear  reaction: 

When  the  gun  is  in  battery  the  tipping  parts 
are  balanced  about  the  trunnion  axis.   This  condition 


267 


implies  that  with  the  gun  in  battery,  the  center  of 
gravity  of  the  tipping  parts  passes  through  the  trunnion 
axis.   When  the  recoil  is  limited  to  a  short  movement 
under  the  breech  when  the  gun  is  fired  at  high  elevations 
the  center  of  gravity  of  the  tipping  parts  is  placed 
forward  if  the  trunnion  axis  and  the  balancing  gear  or 
counterpoise  is  introduced,  balancing  the  weight  of 
the  tipping  parts  about   the  trunnion.   The  trunnion 
reactions  are  modified  by  the  introduction  of  a 
balancing  gear. 

Trunnion  and  elevating  gear  reactions  when  no 
balancing  gear  is  used: 

(a)     During  the'  acceleration  period, 

,Fe+Ks, 
2X=K+Wt  sin  0  *  (— J )  cos  9    (ibs) 


(ibs) 


(b)     During  the  retardation  period, 

Ks+W_x  cos  0 
2X=K+Wtsin  0+( • )cos  9e  (Ibs) 


Ks+W  x  cos  0 

2Y=Wtcos  0-( )  sin  9P   (ibs) 

J 

Ks+W_  X  cos  0 

E  *  (Ibs) 

J 

where  x  =  the  recoil  displacement  out  of  battery. 

Rocker  Reactions: 

T^s  reactions  on  the  rocker  are  primarily  three: 

(1)  The  reaction  of  the  trunnion  upon 
the  rocker,  Xr  and  Yr . 

(2)  The  reaction  of  the  elevating  gear, 
E. 


268 


(3)     The  reaction  of  the  cradle  ,  M, 

and  the  weight  of  the  rocker,    1fr. 
If  k  =  the  perpendicular  distance  fro«  the  trunnions 
to  line  of  action  of  M. 

B  =  the  angle  between  the  line  of  action  of  M  and 

the  "y"  axis. 
h'r=  the  horizontal  distance  to  the  center  of  gravity 

of  the  rocker  from  the  trunnion. 
J  =  the  perpendicular  distance  from  the  trunnion 
axis  to  the  line  of  action  if  the  elevating 
gear  reaction  (i.  e.  equals  the  radius  of 
the  circular  elevating  rack  on  the  rocker). 
Then,  the  cradle  reaction  on  rocker,  becomes, 

Ej-Wrhr    Fe+Ks-Wrhr 

M  =  =  (in  battery)    (Ibs) 

k  k 

Ks-Wrx   cos   0-Wrhr 

=  (out  of  battery)(lbs) 

k 

K» 
approximately  M  =  — 

k 
The  rocker  trunnion  reactions  become, 

2Xr  =  M  sin  B  -W^  sin  tf-E  cos  6e  (Ibs) 

2Yr  =E  sin  9e  -Wr  cos  0-  M  cos  B  (Ibs) 

Layout  of  Balancing  Gear: 

Two  types  of  balancing  gear  have  been  used  ex- 
tensively in  gun  carriage  construction: 

(1)  A  cam  with  chain  type  for  small  field 
mounts . 

(2)  A  direct  acting  balancing  gear. 
For  type  (1),  let 

Wt  =  weight  of  tipping  parts.        (Ibs) 
hj.  =  horizontal  distance  from  the  trunnions  to 
the  center  of  gravity  of  the  tipping  parts 


269 


(gun  in  battery)      (in) 
ro  =  equivalent  radius  of  can  at  horizontal 

elevation  (in) 

rn  =  final  equivalent  radius  of  the  cam  where  the 
cam  arc  has  turned  through  the  maximum 
angle  of  elevation  =  0     (in) 
R  =  niean  radius  of  can.         (in) 
dn=  deflectioa  of  spring  at  zero  elevation  (in) 
dQ=  deflection  of  spring  at  maxim-urn  elevation  (in) 
c  =  spring  constant. 

0=  angle  of  elevation  expressed  in  radius. 
If  ds  -  deflection  of  spring  at  solid  height,  take 
dn  »  (J  to  j)d  solid    ) 

( 
d  =(^  t°  i)  d  solid    ) 


then   _  *tht 
" 


rodn      rndo 


and  dn-d0=( 


To  layout  the  radii  of  cam,  we  have  0  divided  into 

n  parts,  then, 
tht 


Wh  cos 


rtnt 

r  = 


c(dn-r0A0) 
Wh  cos  0 


cos  0 


With  a  balancing  gear  of  this  type,  the  trunnion 
reactions  are  modified  and  now  become, 


270 


if  T   =    the    tension   in   the   chain 

d   =    the   angle  T   oalces   with   the   axis  X(taken   along 
the   axis   of   the   bore) 

Ks+Wrx  cos  0+Fs 

2X=K+"Vfc  sin  0  +  (  -  •  -  )  cos  8Q  -  T  cos  d  (Ibs) 

J 

Ks+¥_x  cos  0+Fs 
=  2Y  =  ¥tcos0-(  jsin  6e  +  T  sin  d   (Ibs) 

J 

The  elevating   gear   reaction   obviously    remains    as  before 
that    is, 


Ks+\frx   cos 
E   =   -  £  -  -  -  (Ibs) 


for  type  (2),  1st 

Tft=  weight  of  tipping  parts     (Ibs) 
ht  =  horizontal  distance  from  the  trunnions  to 
the  center  of  gravity  of  the  tipping  parts 
(gun  in  battery)  (in) 
x^  and  yt  =  coordinates  along  and  normal  to  bore 

from  trunnion  to  canter  of  gravity 
of  tipping  parts  (gun  in  battery) 

0  -  angle  of  elevation. 

£5m  =  max.  elevation 

r  =  radius  from  tbe  trunnion  to  the 

crank  pin  which  connects  the  tipping  parts 
to  the  piston  rod  of  the  oscillating 
cylinder,  (in) 

R  =  reaction  exerted  by  the  balancing  gear  along 
the  piston  rod  of  the  oscillating  cylinder. 
(Ibs) 

dt=  moment  am  of  H  about  trunnion   (in) 

d^=  deflection  of  spring  at  horizontal  elevation 


d]j=  deflection  of  spring  at  maximum  elevation 

(in) 
c  =  spring  constant 

Hj  =  initial  balancing  gear  reaction  (0°  elev.) 
Rt  =  final  balancing  gear  reaction  (0°  elev) 


371 


S  =  stroke  of  piston  in  oscillating  cylinder  (in) 
pt  =  final  air  pressure  in  pneumatic  balancing 

cylinder     (Ibs/sq.in) 
p^  =  initial  air  pressure  in  pneumatic  balancing  cylinder, 

(Ibs/sq. in) 

A  =  effective  area  of  balancing  piston  (sq.in) 
Vo  =  initial  air  volume  (cu.in) 

With  a  metallic  balancing  gear,  the  dimension  of 
the  spring"  may  be  approximated  by  the  solution  of  the 
following  equations: 

=  cos  0_       )  from  nrhich  we  may  obiain  d0,  dv, 


Q, 
(  s  and  c  of  the  spring. 


a  S 


r(l+  cos  — )    ^ 
2     ) 


S  =  2  r  sin  - 


With  a  pneumatic  balancing  gear,  we  have,  for  a  pre- 
liminary approximation, 


Pf 

2Wtxt 

S 

(      Pl      ^      (             ) 

( 

_    —  • 
r  (  1  +c< 

£lDJi 

\               )      (.CM  .  in  / 

) 

3S2 

1  

( 

Pi 

) 

S  =    2r   sin 

*• 

f.    .      Pf 
(in;     —  =   cos   0. 

( 

2 

Pi 

(approx) 

) 

With  a  direct  acting  balancing  gear,  the  trunnion  re 
actions  are  modified  and  become, 


Z72 


if 

R  =  balancing   gear  reaction      (Ibs) 
qr-    angle  between  R  and  y  axis 

dt   =    moment    arm   of  R  about   the    trunnion   at   any 

elevation  0  (in) 

when   the   recoiling   parts   are   in  battery: 

2X«K+Wtsin  0+E   cos   8e    +  R  sin   9r        (Ibs) 
2Y=Hft   cos  J0+E   sin  9e  -R  cos  6r  (Ibs) 

Wtxt   cos    0        21*txt   cos    ^ 
R   »  : =   a (Ibs) 


Ks  +  P^e 


when  the  recoiling  parts  are  out  of  battery  :- 

2X»K+R  sin  6r+E  cos  6e+Wt  sin  0    (Ibs)         ) 

( 

2Y»Wt   ces   U   +  E   sin  6ft-R  cos   er          (Ibs)  ) 

( 
2Wtxfc    cos    t 

R  =  . (roughly)  (Ibs)  ( 

r(l+cos— -)  ) 

2  ( 

Ks+W_   x   cos   0 

Is (Ibs)  ( 

J  ) 

It   is   evident    that    th«   elevating   gear   reaction 
remains    the   same   with  or   without    a  "balancing   gear 
while    the    trunnion  r«actions   are   modified  both  by 
the  position  and  Magnitude   ef   the  balancing   reaction. 


273 


Strength  of  the  trunnions 

The  critical  section  ef  the  truT>',i«»s  is  usually 
where  the  trunnion  joins  the  cradle.   L«t,  "«n*  represent 
this  section.  [See  fig.  (9)]. 

a  =  distance  fro»  "mn11  to  center  of  top  carriage 

bearing  . 

b  *  distance  from  "mn"  to  center  of  rocker  "bearing 
MX  =  the  bending  moment  at  "mn"  in  the  plane  of 

the  X  component  reactions. 
My=  the  bending  moment  at  "mn"  in  the  plane  ef  the 

Y,  component  reactions. 

M  «  the  resultant  tending  moment  on  section  "an". 
f  =  aax.  fibre  stress   (Ibe/sq  .  i'ff) 

D  =  distance  ef  fhe  trumnien  at  section  "mn" 
then 


Mx=  X.a+Xrb  (in  Ibs)  and  M  = 
My=  Y*+Yrb  (in.  Ibs) 


hence 


/10.18  M 
D  =  / (in) 


Stresses  in  cradle  or  recuperator  forging: 
Let 

Ql  and  QZ  =  the  front  and  rear  normal  clip  re- 
actions . 

xt  and  xg  ~  the  x"  coordinates  of  these  re- 
actions with  respect  to  the 
trunnions. 

dx  and  da  =  the  distance  of  the  friction  co»- 
ponents  ef  Q±  and  QZ  from  the 
neutral  axis . 

B  =  the  resultant  of  the  braking  pulls  re- 
acting on  the  cradle. 
d-=  the  distance  from  the  neutral  axis  to  "B". 


274 


It  =  moment  of  inertia  of  a  cross  section  at 

the  trunnions. 
yt  =  distance  of  extreme  fibre  from  neutral  axis 

at  trunnion  section. 
ft=  fibre  stress  due  to  bending  and  direct  pull 

or  thrust  at  the  trunnion  section. 
Ic  =  nonent  of  inertia  of  a  cross  section  at 
the  point  of  contact  of  the  elevating 
arc  with  cradle. 

AC  =  area  of  cross  section,  at  the  point  of  con- 
tact of  elevating  arc  with  cradle, 
y  =  distance  to  extreme  fibre  from  neutral  axis 

of  elevating  arc  section. 
fc  =  fibre  stress  due  to  bending  and  direct  pull 

or  thrust  at  the  elevating  arc  section. 
A^  =   area  of  a  cross  section  at  the  trunnion, 
then 


)yt   UQ- 

+  — • —    for  the  braking  reaction 


in  the  rear, 


ft  =  — - — i— +  — i for  the  braking 

^     reaction  in  the 
front . 


*x, 

*      x,  •   ""  ~^«*~  •"  w        j^  •  UVA 

for  the  brak- 
ing reaction 
in  the  rear. 


U°2 

f   =   "   '   " = ^^  +  — i-      for  the  bralc- 

T  A 

Ac  c      ing  reaction 

in  the  front. 


-  -Jfc'^SJ  APPENDIX----- 
APPENDIX  CHAPTER  IV-  INTERNAL  REACTIONS. 

BKACTIOHS  AMD  STRESSES  IKDPCSD  IK  ELEVATING   AMD  TR«VERS- 
IH8  MECHANISMS: 

STRESSES  DUE  TO         The  reaction  exerted  on  the 
FIRING.  elevating  mechanism  due  to 

firing  equals, 

In  Battery,  Out  of  Battery 

Fe  +  Xs  ?%v   Ks*Wrx  cos  J0 

J  cos  20  J  cos  20 

where 

F  =  max.  powder  force 

K  =  Total  resistance  to  recoil 

YT  =  weight  of  recoiling  parts, 
r 

x  =  displacement  out  of  battery. 

J  =  radius  to  pitch  line  of  elevating  arc  fron 

center  of  trunnions, 
e  =  J_  distance  fron  axleof  core  to  center  of  gravity 

of  recoiling  parts. 
S  a  J_  distance  from  line  parallel  to  axis  of 

gun  through  center  of  gravity  of  recoil- 
ing parts  to  center  of  trunnions. 
It  is  highly  desirable  to  reduce  the  reaction 
E,  since  it  stresses  the  teeth  of  the  elevating 
mechanism.  To  reduce  this,  we  may, 

(1)  decrease  "e"  by  so  distributing  the 
mass  of  the  recoiling  parts  as  to  bring 
its  masses  ss  near  coincident  with  the 
axis  of  the  bore  as  possible. 

(2)  decrease  "S"  by  bringing  the  trunnion 
axis  along  a  line  through  the  center  of 
gravity  of  the  recoiling  parts  and 
parallel  to  the  axis  of  the  bore. 

(3)  increase  "J"  whenever  feasible  in 
a  construction  layout. 


275 


276 

In  certain  types  of  oounts  as  those  contain- 
ing a  recoiling  cylinder,  the  piston  and  rods  "being 
fixed  to  cradle,  the  center  of  gravity  of  the 
recoiling  parts  is  necessarily  considerahly  lowered 
froa  the  axis  of  the  bore  and  therefore  "e "  is  in- 
herently large.   With  large  mounts,  counterweights  or 
bob  weights  are  sonetines  introduced  to  decrease  "e". 
In  this  type  of  mount  without  a  counterweight  or  "bob 
•eight  a  friction  clutch  or  hand  brake  are  often  in- 
troduced on  the  elevating  gear  shaft  or  adjacent  gear 
shaft.   Then  E  becomes  limited  to  that  required  to 
overcome  the  friction  of  the  clutch  or  brake  and  a 
large  reaction  on  the  elevating  mechanism  is  thus  re- 
duced. 

With  a  cone  clutch,  we  have, 

uPr 

E  =  : — -  ,  where  P=  total  spring  load. 

r  =  mean  radius  of  clutch 
re  =  pitch  radius  of  gear  or 

pinion, 
n  -  coefficient  of  friction  = 

0.15  approx. 
2«  =  cone  angle 

With  a  dislc  clutch,  we  have, 

),  where  P  =  total  spring  load. 

r2=  outer  radius:  rt=  inner 

radius  of  dislc. 
k  =  total  no.  of  friction 

surfaces . 
n  =  coefficient  of  friction 

-  0.15  approx. 

FRICTION  OF  TRUNNIONS     Tn  elevating,  or  traversing 
AND  TRAVERSING  PIVOTS.   a  gun,  a  large  amount  of  the 

energy  needed  is  that  required 
to  overcone  the  friction  of 
the  pivot  about  which  the  gun  is  traversed. 


277 


Trunnion  friction: 

During  the  elevating  process  the  load  on  trunnions 
equals  the  weight  of  the  tipping  parts,  when  the 
trunnion  is  sufficiently  free  from  binding,  the  con- 
tact is  along  a  narrow  strip. 

Then  u 

t 
nR  sin  0+R  cos  9  = -r—     )  u=  coefficient  of 

where  tan  0  *  n  lO   friction. 

Wt   )  R  =  »or»*l  pr«s- 

.  *  .R  (sin  0  tan  £J+  cos  0)=~r —  (     SUP« 

)  r  =  rsdius  of  trwa- 
(     nion. 


and  the  friction  moment 

Mt=  R  tan  0  .r  =  —  r  sin  0 

m 

Since  0  is  small,  tan  0  =  sin  t  approx. 
hence       W  Wt 

M+  =  n — —r   =  0.15 — •—  r  approx. 
$o  l      2  2 

In  starting  n  nay  be  as  great  as  0.25  an&  proper 
allowance  should  be  nade. 

Since  the  load  brought  on  the  trunnions 
during  firing  is  greatly  in  excess  of  that  on 
elevating  the  gun,  the  bearing  contact  may  be 
divided,  one  part  to  carry  the  major  of  the 
firing  load  and  the  other  to  carry  merely  the 
weight  of  the  tipping  parts.   This  is  ac- 
complished constructively  by  allowing  play  in 
the  bearing  which  sustains  the  firing  load,  and 
holding  the  tipping  parts  for  elevating  or 
transportation  merely  on  a  spring  cushion, 
the  reaction  of  the  spring,  for  a  deflection 
just  sufficient  to  lift  the  tipping  parts  just 
clear  from  the  firing  bearing,  being  equal  to 
the  weight  of  the  tipping  parts.   Thus  it  is 
possible  to  reduce  the  friction  by  using  a 


278 


smaller  trunnion  diameter,  in  tliat  part  of  the  bearing 
that  is  spring  borne  since  the  bearing  surface  for  a 

nominal  bearing  pressure  can  be  greatly  reduced. 
Pivot  friction  in  traversing: 


This  friction  will  vary  considerably  according 
to  the  type  of  bearing  used.   We  will  consider  three 
types  of  pivots,  1*  flat  circular  pivot,  2°  flat 
hollow  circular  pivot,  and  3°  conical  pivot.   To  esti- 
mate the  load  brought  on  the  pivot,  let, 

Va  =  pivot  reaction  or  load (vertical ) 

V^  =  normal  load  of  traversing  guides  (vertical) 

Wt  =  weight  of  tipping  parts. 

1  =  horiiontal  distance  between  Va  and  Vv 

d  U 

l+=  horizontal  distance  from  W+  to  Vv 

v  u  U 

Wc  =  weight  of  top  carriage. 

lc  =  horizontal  distance  from  ¥t  to  V^ 

then      ff  1  +W  1 

Va  =  -  load  on  pivot  during  traversing. 

If  Kt=  the  friction  couple  exerted  at  the  pivot 
during  the  process  of  traversing  we  have  for  the  various 
types  of  bearings, 

1°  for  flat  circular  pivot: 

The  friction  on  an  elementary  zone  = 2  re  r  dr 

The  moment  of  this  friction  about  the  center  =  — j2nr*dr 

L     27a   ro   2     2Vanro 
The  total  friction  =  — <g —  n/   r  dr  =  — 

ro   o  3 

Therefore  for  a  flat  circular  pivot,  letting  n  - 
0.15, 

Mt  »  0.1  Va  r0 


279 


2°  for  flat  hollow  circular  pivot 


The  total  friction  evidently  becomes, 
—  I  *   r'dr  = 


2Van     r          2Van    r'-r' 


3 
hence,    letting   n  =   0.15 


3°for  conical  pivot: 

The  intensity  of  vertical  pressure  on  the  projected 
area  of  the  bearing  =    y 


n(rf-r*>     V 

If  the  cone  makes  an  angle  2«,  and  pn  equals  the  in- 

tensity of  normal  pressure,  then, 

rd6dr  rd9dr 

the  normal  pressure  on  area  —  —  =  p_  —  . 

since  B  fJin  «t 

the  vertical  component  of  this  pressure  = 
rdedr 


rd  6dr 


sina 

but  the  pressure  on  the  projected  area  rd6dr  =  p  rd9dr 
hence  p¥  =  Pn  =    ?. 


2nrdr 


the  friction  on  a  differential  zone  =  n 


2 

the  total  friction  moment,  therefore  becomes, 


If  then  we  let  n  =  0.15 


280 


VELOCITY  RATIOS  OF  ELEVATING     Elevating  and  travers- 
AKD  TRAVERSING  MECHANISMS     ing  mechanism  consists 

usually  of  a  train  of 
spur,  bevel,  helical 
screw  and  worn  gears. 
l°)-Velocity  Ratio  of  spur  gear: 

Since  *trt  =  v»tra    )  *  =  angular  velocity 

^  r  =  radius  to  pitch  line. 

we  nave  -±  =  — L  -  _L     (  n  =  no.  of  teeth. 


2°)-Velocity  Ratio  of  Bevel  gears: 


Again  w  r  =  w  r  where  r   and  r   are  the  outside 

1   t      Z   2  1          2 

radii  of  the  gears: 

The  angle  of  coning  for  the  first  gear,  equals, 


r. 


"  * 

tan  6t   «—  (  6t  =  1  angle  of  cone,  ) 

or  the  second  gear 

tan  62  =  —  (62  =  -  angle  of  cone) 


hence  w 

1  2 

1    =  tan  9.  and  -  =  tan  0 
w 


Therefore  we  may  take  any  two  common  radii  in  ob- 
taining tlie  velocity  ratios,  again 


3°)-Helical  screw  gears: 

Tfe  have  for  the  velocity  of  the  common  normal, 


w  r  cos 


r2cos  9, 
r  cos  9 


but  also, 


Pn  = 
then 


=  pcos 


cos  9. 


Pn 


cos  9. 


281 


)  0t  =  angle  be- 
l*(F*     tween  axis 
)      of  gear  fl 
(      and  perpend- 
icular to 

(      common  normal. 

.  )  98  =  angle  between 
(      axis  of  gear 
)      #2  and  per- 
(      pendicular  to 
)      common  normal. 

(  pn  =  common  normal 

)      pitch. 

(  pt  =  circuraferent- 

)      ial  pitch  gear 

(      #1. 

)  n  =  circumferent- 


tial  pitch  gear  #2. 

n   =  no.  of  teeth  gear  fl. 

n2  =  no.  of  teeth  gear  f2, 


Hence  —  =  —  = 


r  cos9 


r  cos9 
i     1 


If  0  =  the  total  angle  between  the  axis  of  the  gears 

in  mesh,  then 
since  p   =  p   cos  Q   =  p  cos  9 


cos  9t  = 


cos  e  =  "•  si" 


~fP2~2  PtP2cos 


therefore 


282 


Further  the  axial  pitches,  become, 

TBX  =  pt  cot  6t   and  m2  =  pa  cot  9f 

4°)"Velocitv  Ratio  Worm  gears: 

Though  a  worm  gear  is  a  specified  type  of  nelical 
screw  gear  when  Si  =  90°.  it  is  convenient  to  consider 
this  type  as  a  separate  classification. 

When  0  =  90°, 


cos  6.  * 


=  sin  9 


therefore  the  axial  pitch  of  one  equals  the  cir- 
cumferential pitch  of  the  other. 

The  worm  of  a  worm  gear  has  one  to  two  or  three 
threads  while  the  gear  has  many  threads. 

Now,  for  a  single  thread  worm, 


rwsin  9 
r.,  cos9 


-  —  tan  9 


Directly,  we  have. 


% 

P^T 


but 


2n 


=  tan9.w»r 


wlw 


;  *g=  ang.  velocity 
(     of  gear  wheel . 

)  ww-  ang.  vel.  of 
(     worm  wheel. 

)  rg  -  pitch  radius 

(     gear. 

)  rw=  pitch  radius 

(     of  worm 

)  p  =  axial  pitch  of 

(     worm 

)  9  =  angle  of  helix. 


—  =  —  tan  9 


Thus  the  ratio  of  angular  velocities  depends  upon 
the  angle  of  the  helix  of  the  worm. 
With  a  "n  "  threaded  worm, 


:  nwP  — 
£n 


g     r 
and  —  =  n  -^ 

ww     rg 


tan  6 


283 


p 
In  terms  of  the  number  of  teeth,  since  =  tan  9 


w 


p          nw 

n...  ian  9  -  — 

2nr  ng 


and  for  a  single  threaded  worm,  since  nw  =  1 

*  - 


Velocity  ratio  in  gear  trains: 

Combining  the  previous  equations  from  one  pair 
of  elements  to  the  adjacent  pair,  we  finally  arrive 
at  the  velocity  ratios  of  the  first  and  last  wheels 
of  the  trains  in  terms  of  the  number  of  teeth  or  radii 
of  pitch  circles:   In  this  combination,  it  is  always 
preferable  to  set  the  general  equation  up  in  terms  of 
the  number  of  teeth  rather  than  the  radii  of  pitch 
circles,  for  then  the  relations  are  independent  of 
the  type  of  gearing  and  velocity  ratios  between  a 
meshing  pair  are  inversely  as  the  number  of  teeth  or 
threads. 

Thus  assume  worm  #1  to  drive  worm  gear  #2,  while 
bevel  gear   #3  on  same  shaft  as  gear   #2,  drives 
bevel  gear  #4,  then  helical  screw  gear  #5  on  same 
shaft  as  gear  #4,  drives  helical  screw  gear  #6  and 
finally  gear  #7  on  gear  shaft  #6,  drives  pinion  #8. 

Since  2  and  3,  4  and  5,  and  6  and  7  are  on  same 
shafts,  we  have, 

then, 


wwwwnn          nn 

13  5794  88 

hence  —   *   —    *  —    *  —  =  —   *   —   *   —   *  — 


284 


wt   nt   n4   n    n 
therefore  —  »  —  x  —  x  —  x  — 
w    n    n    n    n 

8       1       3       5       7 

If  Tt  =»  torque  on  worm  shaft  #1  and  T   the  torque 
on  pinion  shaft  and  e  the  efficiency  of 
the  total  gearing,  then  T§wa  =  e  T^ 

hence     -p   n  xn  xn  Mn 

Tt  *  —  ( )  where  Tt  =  required  power 

torque  and 
T$s  load  torque  at 

end  of  train. 

REACTION  BETWEEN  GEAR  PAIRS:-     The  efficiency  of 
EFFICIENCY.  spur  and  bevel  gears 

is  hifh  compared  with 
helical  screw  gearing, 
especially  of  the  worn 

gear  type.   The  very  large  force  and  velocity  ratio 
attainable  by  the  latter  makes  this  type  preferable. 

1°  Spur  Gears: 

For  approximate  calculations,  the  normal  reaction 
between  'the  teeth  will  be  taken  at  an  angle  of  20*  with 
the  tangent  to  the  pitch  circles.  The  effect  of 
friction  between  the  teeth  is  to  cause   the  resultant 
reaction  to  make  an  angle  of  25°  with  the  tangent  to 
the  pitch  circles. 

Therefore  if  T  is  the  torque  to  be  transmitted, 
the  reaction  between  the  teeth  R,  becomes, 

T    x   12 

where  T   is   measured   in    (Ib.ft) 


r  cos  25°  , .   . 

r  is  neasured  in  (in.) 

Ifhen  smoother  running  is  required  with  high 
velocity  ratios  helical  spur  gears  have  been  extensive- 
ly introduced.   If  B  =  the  angle  between  the  normal 
to  a  tooth  surface  and  the  tangent  to  the  circumference 
(i.  e.  normal  to  axis  of  rotation),  then 


285 


T  *  12 
r  cos  25  cos  B 

If  b  =  tooth  rim  breadth,  the  mean  pressure  is  dis- 
tributed along  a  linear  element  -  b  sec  6  and 

therefore  the  pressure  on  an  element  becomes 
per  linear  inch,  proportional  to 

T  *  12 


r  cos  25. b   the  same  as  in  ordinary  spar 
gearing. 

2*  Bevel  Gears: 


The  reaction  between  bevel  gears  takes  place  at 
the  intersection  of  the  common  pitch  circles  of  the 
cone  elements  of  the  gears,  and  this  intersection  is 
in  the  plane  of  the  axis  of  the  gearing.   The  neutral 
reaction  between  the  teeth  makes  an  angle  approximately 
equal  to  20°  with  the  normal  to  this  plane  due  to  the 
contour  of  the  tooth.   The  tangential  component  pro- 
duces no  axial  thrust.   The  component  parallel  to  the 
plane  =  P  tan  20,  where  P  is  the  tangential  component. 
This  component  is  also  perpendicular  to  the  common 
intersecting  line  of  the  two  cones.   If  the  cone  angle 
of  gear  #1  equals  28  then  the  cone  angle  of  gear  #2  = 


The  axial  thrust  for  gear  fl  becomes,  P  tan  20°  sin6 
The  axial  thrust  for  gear  *2  becomes,  P  tan  20°  cos6 
Further  the  radial  reaction  "between  the  teeth  and  there- 
fore the  radial  bearing  loads  for  gear  #1  and  gear  #2, 
becomes, 


R'  =  /p2  4.  (p  tan  20°  cos8)2  =  P  /l  +  (tan  20°  cose)' 

Where    T  x  12 

P  =  ;  and  28  -  the  cone  angle  of  gear  #1 

r 

71 

2(-  -  8)=  the  cone  angle  of 
gear  #2. 


286 


3°  Helical  Screw  Gears: 

Assuming  the  axis  of  the  gears  to  make  an  oblique 
angle  £J t  the  angle  6  between  the  contact  line  of  the 
teeth  and  axis  of  gear  #1  is  given  by  the  expression 

P2sin  0 
cos  9t  = 

while  the  angle  9  between  the  contact  line  of  the 
teeth  and  axis  of  gear  #2,  is  given  by  the  expression 

p  sin  0 
cos  9.  =  ,. .  * .. 


where  pt  and  pz  are  the  respective  circumferential 
pitches  of  the  two  gears. 

The  reaction  between  the  teeth  makes  a  resultant 

angle  i  with  the  normal  to  the  contact  line,  where 
tan  i  =  n  the  coefficient  of  friction 

Then,  if  Tt  is  the  external  torque  exerted  on 
gear  #1,  we  have  T±  =  R  cos  (9t  -  i).r± 
while  if  Tg  is  the  torque  on  gear  #2,  Ta=R  cos(92+i).r. 
Work  expended  =  T  w 


w   '  r  cose 
Work  delivered  =  T8wt 
Then  the  efficiency  E  becomes, 

T  w    cos(9  +i)cos9 
E  =  -2-2- ? - 


tt   cos(0t-i)cos92 
The  reaction  on  the  teeth  is  given  by 

Tt 
r.cos(9  -i) 


T   sin(6  -i) 
R  sin(et-i)= ^ 


287 


and  the  thrust  along  gear  shaft  f2,  is 


t 
R  sin(8   +  i  )  =  — 


r  cos(9t-i) 

The    total    radial   bearing    load   for   shaft   of  gear   #1 
balances, 

T     cos(6  -i)        T 
R  cos(9   -i)=   -^  _— i-— =   -1 
rt   cos@f-i;  rx 

and   the   total   bearing    load   of  gear   shaft   #2 
balances,  ^ 

R.  cos(9    +i)=  — - 


4°  Worm  Gear: 


Though,  worm  gearing  is  a  special  case  of  3°,  a 
separate  analysis  will  be  made  due  to  the  greater  use 
of  this  type  of  gearing  as  compared  with  helical 
gearing  when  the  shafts  are  not  at  right  angles. 

Let  xx   and  yy*  be  the  coordinate  axis  along 
and  perpendicular  to  the  axis  of  the  worn  in  the 
plane  perpendicular  to  the  radius  of  the  pitch  line  of 
the  worm  through  the  common  pitch  point  as  origin. 

Let  S  =  the  angle  that  the  contour  of  the  tooth 
makes  with  the  normal  to  the  xy  plane  at  the  pitch 
point,  and  6  =  the  angle  of  helix. 

Let  R  =  normal  component  between  worm  and  gear 
tooth, 

nR  »  friction  component  between  worm  and  gear 

tooth, 
then  the  axial  thrust  along  worjn  wheel  is 

X  -  R  cos  S  cos  9  -  nR  sin  6  and  the  turning 
component  on  the  worm  is 

Y  =  R  cos  S*  sin  6  +  nR  cos  0 

and  the  thrust  tending  to  separate  the  teeth  is 
Z  =  R  sin  S  . 

It  is  to  be  noted  that  tan  S  =  tan  S  cos  6 


288 


If  T_3  torque  applied  to  worm  gear 

Tg=»  torque  on  gear  wheel 
then, 

Tw=Yrw   and  Tg=Xr^     rw=  radius  of  worm  gear 

rg=  radius  of  gear  wheel 
To  determine  the  efficiency,  .vs  hava 

but  —  =  —  tan  0 


then  „! 

cos   S  cos0-n   sin8  . 

e   =  — — — — — — — — — —       tan   6 

cos   S'sin0+n   cos6 

n  tan0 


cosS1 

-)  tan  0 


n 
cosS1 

tan  e  n 

e=  ; — ~~  where  k  =  tan"1— — — - 

tan(e+k)  cos  S' 

and  tanS'-tanS  cos  6 

COMBINING  THE  REACTIONS     In  gear  transmissionhaweA 
FROM  ONE  PAIR  TO  ANOTHER,  between  two  elements,  #1 

and  «2, 


f       m  w  =  angular  velocity 

hence  ;r-  =  8t  —  —   Likewise  between  gear  elements 
Ti      wt 

#3  and  #4,  -p       w 


Then  if    gear  #2  is   on  same   shaft   as  gear   |3,   we 

have  T     =  T  and   w     =  w        hence 

23  ?                  3 

T          T  w           w 

42  's-1 

-    x    —  =     6       —    E       —  — 

T3            Tl  *4^4 


=  e  e   — 

ia  34 


289 


w 

Now  the  velocity  ratio  —  may  be  obtained  as  outlined 

w 

in  previous  discussipn  on  velocity  ratios. 

In  the  proceeding  discussion  the  inertia  effect 
of  the  gear  elements  has  been  neglected  in  comparison 
with  the  friction  developed  between  the  gears. 

TORQUE  AND  POWER  REQUIREMENTS     In  elevating  ertravers- 
FOR  ELEVATING  AND  TRAVERSING    ing  a  gun,  we  nave  three 
MECHANISMS.  important  periods  :-(a) 

accelerating  period, 
(b)  the  period  of  uniform 

motion  and  (c)  the  retardation  period.  The  maximum 
torque  obviously  occurs  during  the  acceleration  and 
power  is  continued  through  period  (b),  while  the 
friction  of  the  mechanism  brings  the  system  to  rest 

during  period  (c). 

Let  1^  =  moment  of  inertia  about  the  trunnions  of 

the  tipping  parts. 
1^  =  moment  of  inertia  about  the  vertical 

traversing  pivot  of  the  tipping  parts 

and  top  carriage. 

E  =  elevating  gear  (tangential  reaction.) 
J  -  radius  of  elevating  arc. 
r  =  radius  of  traversing  arc. 
Mt~  friction  moment  of  trunnions 

M^=  friction  moment  of  traversing  pivot. 
Then  during  the  acceleration, 


J  t  t  dt«      for  eievating  the  gun 

E.r-M'=l'  ill 

fc   *   dt     for  traversing  the  gun 

Now  MI  and  M^  are  constant  depending  approximately 
on  the  weight  on  the  bearing,  while  on  the  other  hand 
E  and  E'  depends  on  the  elevating  or  traversing 
motor  characteristics. 


290 


Neglecting  the  inertia  of  the  gear  elements,  we 
have,  the  torque  transmitted  varying  directly  as  the 
number  of  teeth,  that  is  between  any  two  gear  elements, 

T         i 

for  gear  pair  1-2 


3 
—  =  --  -       for  gear  pair  3-4 

Ten 

4       34    « 


for  gear  pair  7-8 


If  gears  2  and  3,  4  and  5,  6  and  7  are  assuned  on 
sane  respective  shafts, 

T,  =  T3,  T4=  Ts,  T6=  T7 

then 

Ii  .  L  .  L.  ,  !i  .  L  =  A  „  i.  ,  L.  „  ^i,J_  .  ^_  , 

T.     T,     T.     T.     T.     °2     "4     ".     ",  Si     '.4 


e     t 

5  •       7t 

Now  TV  =  E  re  and =  *  *  *  

e    e     e     e     e 

t  t       34        S  8        78 

then      gr    n.    n    n    n 

T   =  — -  (  — -  x  -i  x  -5.  x  — 1) 
«     na    n4    ne    "« 

hence   T    „    n    n    n         d20 

e  — —  ( x  — i  x  — i  x  — S)J-M 4.  =  !+  f°r  elevating 

re    ni    na    n,    nr         dt 

the  gun. 


291 


l          2  4  s  «  II 

e  —  (  —  *  —  *  —  *  —  )r-M*  =  It  —  -  for  traversing 

'   *   z 

the  gun. 


dtz 


and  for  the  sngular  velocity  ratios,  we  have, 


J 
and  w  =  —  wt;  -  for  spur  or  bevel  gears: (elevating; 


r    i 
*  =  —  *t:~  ^or  3Pur  or  tevel  gears:  (traversing) 

re 


=  -2—  wt  :  for  worm  gear  in  contact  with 
e    np 

elevatin  arc   (elevating) 

2nr 

.  =  -TP—  wt  :  for  worn  gear  in  contact  with 
•   n  p    •• 

traversing  arc    (traversing; 


CHAPTER   V. 
RECOIL  HYDRODYNAMICS. 

OBJECT.  The  modern  recoil  system  is 

essentially  a  hydropneunatic  device 
for  dissipating  the  energy  of  recoil 
by  so  called  hydraulic  throttling 
losses,  and  returning  by  means  of  the 
potential  energy  stored  up  in.  the  compression  of  air, 
the  recoiling  mass  into  battery.   The  potential 
energy  at  the  end  of  recoil  required  to  return  the 
piece  into  battery  is  relatively  small  compared  vritb 
the  energy  dissipated  by  the  hydraulic  braking. 
Further  the  potential  energy  of  counter  recoil  is  in 
greater  part  dissipated  by  the  hydraulic  counter  re- 
coil buffer  in  the  return  of  the  recoiling  mass  into 
battery. 

In  the  design  of  the  braking  system  misunder- 
standing may  result  due  to  incomplete  comprehension 
of  the  fundamental  principles  underlying  the  hydraulic 
throttling  and  the  various  hydraulic  reactions.  Hence, 
in  this  chapter  a  resume  of  the  essential  principles 
underlying  the  hydraulic  phase  of  recoil  design  will 
be  attempted. 

ELEMENTARY  HYDRAULIC  Consider  an  ordinary  tension 
BRAKE  brake  (fig.l)  the  oil  being 

throttled  through  apertures  in 
the  brake  cylinder  from  the 
front  or  rod  side  of  the  piston 
to  its  rear. 

Let  ax  =  area  of  the  variable  apertures  or 

orifice . 

An  =  effective  area  of  piston  on  rod  side. 
A  =   total  area  of  cylinder. 
ar  =  area  of  rod. 
Pn  *  total  hydraulic  pull. 

293 


294 


u 


Dp 

L 


J 


CvJ 

00 

L 


295 


Vx=velocity  of  recoil  at  displacement  x. 
vx=  velocity  of  oil  through  apertures. 
D  =  weight  of  fluid  per  unit  volume^ 
p  =  p^=  intensity  of  hydraulic  pressure. 
C  =  contraction  coefficient  of  orifice. 
K  =  reciprocal  of  contraction  coefficient. 
For  a  displacement  dx,  the  mass  of  liquid  moved 
by  the  displacement  of  the  piston,  becomes, 

D  Ah  dx 

and  due  to  the  contraction  of  the  liquid 

g 

in  the  throttling  aperture  or  orifice,  its 

effective  area  is  reduced  to  C  ax,  therefore,  the 
mass  is  accelerated  to  a  velocity 

A      v          \r    A      w 

Hh    vx             Hh    Yx  1 

vx  = =  ,    since   K  =  -5—     now   the  energy 

SLy  £L« 

of  the  jet, 

*  D  Ah  dx   . 


vx  becomes,  dissipated  by  a 

loss  due  to  sudden  expansion 

in  fhe  rear  part  of  tlie  cylinder,  where  we  find  a  void 
equal  to:  (A-Aj1)x=  arx  .   By  the  principle  of  virtual 

work,  evidently        x  A,  v 

-  D  Ah  dx  *h  vx 

p*  d«-  -J^-  (T^-> 

hence    1  D  Au  V* 


g  c  a  x 

that  is  in  terms  of  the  liquid  pressure 
:  D  K«  A>  V 


Consider  again  a  brake  where  the  throttling 
occurs  between  the  hydraulic  cylinder  A  and  a  re- 
cuperator cylinder  B  containing  a  floating  piston 
which  is  contact  with  the  oil  on  one  side  and  the 
air  on  the  other.   See  fig.  (2). 

Let  p  =  pressure  intensity  against  hydraulic 
or  recoil  piston. 


296 


Aj,  =  effective  area  of  hydraulic  piston. 

ax  =  throttling  area  between  the  two  cylinders 
which  we  may  assume  is  controlled  by  a 

spring. 

vx  =  velocity  through  orifice. 
Vx  =  velocity  of  recoil. 
Va  =  velocity  of  floating  piston. 
Aa  =  area  of  floating  piston. 
pa  =  pressure  intensity  against  floating  pia- 

ton. 

x  =  displacement  of  floating  piston. 

Then  by  the  law  of  continuity,  A^  dx  =  Aa  dx 
Due  to  the  contraction  and  sudden  expansion  of  the 
liquid  from  the  throttling  apertures,  the  loss  due 
to  eddy  currents  becomes, 

D  A   dx   «h  "x 


By  the  principle  of  virtual  work,  we  have, 

r  D  Ah  dx    Ah  Vx  „ 

,  ,1      2       i)         /   U    A  \l 

Ph  Ah  dx  -paAadx   =  -  (  -  ) 

g       C  ax 

Neglecting  the  slight  change  in  the  total  kinetic 
energy  of  the  liquid  in  its  virtual  displacement. 
Simplifying,  we  obtain, 

p  K'    v 


g  a  x 

which  gives  the  drop  in  pressure  through  the  orifice, 
or  the  so  called  throttling  drop,  Obviously,  Pn=Ph*h» 


as  before,    i 

-  D  K*AV 


(4) 


PRINCIPLES  OF       (1)     Though  in  the  analysis 
HYDRODYNAMICS.    of  recoil  brakes,  liquid  viscosity 

is  an  item  of  importance,  the 

viscosity  effect  in  modifying  pressures  is,  with  a 
few  exceptions,  small,  and  therefore,  for  a  first 


297 


approximation  we  will  consider  an  ideal  fluid,  that  is 
a  liquid  with  no  viscosity. 

(2)  It  may  be  shown  by  simple  analysis  in  the 
consideration  of  a  small  tetrahedron  or  triangular 
prism  that  the  pressure  intensity  on  all  planes  at  a 
given  point  within  a  fluid  is  the  same,  the  bodily 
forces  such  as  gravity,  inertia  resistance  etc.  in 
limit  being  eliminated  since  they  are  functions  of  high- 
er order  (three  dimensions)  than  the  surface  pressures 
(two  dimensions). 

(3)  By  higher  analysis  it  may  be  shown  that 
fluids  flow  in  so  called  stream  lines  and  therefore 
the  variation  of  pressure  with  velocity  at  various 
points  along  the  stream  line  as  well  as  the  change 
in  such  due  to  the  acceleration  of  the  fluid  as  a 
whole,  may  be  determined  by  a  consideration  of  the 
pressures  on  continuous  differential  elements.  Due 

to  the  mutual  action  between  differential  elements,  we 
nay,  by  simple  integration  along  a  stream  line  determine 
the  pressures  at  the  extremities  of  a  stream  line  tube, 
that  is  the  end  pressures  as  well  as  the  terminal 
velocities. 

Consider  a  differential  element  A  8  C  D  along  a 
stream  line,  of  cross  section  w  of  length  ds  and  a 
circumferential  perimeter  c. 

Let,  the  intensity  of  pressure  on  A  D  be  p,  the 
weight  per  unit  volume  be  G,  then  for  the  pressures  on 
the  surface  A  B  -  C  D  and  the  wall  of  the  tube,  we 

have 

dp 

pw-(p+  —  ,ds)(w-dw)-pcds  sin  <*-D  "  ds  sin  J0  = 
ds 

—  but  cds  sin  <x  =  dw.   Simplifying  and 

g   dt 

dividing  through  by  w,  we  have,   -  dp-D  ds  sin  0  - 

2*1*1  (5) 

g   dt 

dv  s    dv          dv   dv     dv 
but  dv  =  —  dt  +  —  ds   hence  —  =  —  +  v  -—  -which 
dt     ds          dt   dt     ds 

shows  the  acceleration  is  both  a  time  and  space 


298 


function,  inserting  in  (5)  we  obtain, 

Dds  dv     dv 

-dp-D  ds  sin  0  (77  *  *  7~>  <6) 

g   dt     ds 

Integrating  from  (1)  to  (2)  along  a  stream  line,  since 
the  mutual  reactions  between  contiguous  particles  can- 
eel  out,  we  have, 


;  i  'd°<zi>*  4?1 


1    ,dv   v"v 


Obviously, 

/  *  ds  sin  0  =Z  -Z 


hence 


dv 

The  term  /  ds  —  is  of  special  interest  and  when  it 
dt 

occurs  the  motion  is  not  steady.   This  tern  is 

theoretical,  always  existing  in  a  recoil  brake,  since 
the  fluid  in  addition  to  a  space  variation  of  velocity 
due  to  changes  of  sections,  is  on  the  whole  ac- 
celerated as  well. 

.  ,  dv  dv 

To  evaluate  /  ds  —  it  is  necessary  to  express  -— 

dt  dt 

as  a  function  of  s.   If  now  we  assume  the  same  stream 
lines  to  exist  whether  accelerated  or  under  uniform 
steady  motion,  we  have, by  the  equation,  of  continuity, 

wiVi  =  *2V2=  *3  V3    and 

dvt     dv^     dva 

1  dt      dt    '  dt 
hence  knowing  the  acceleration  at  one  section, 

dvn   w!  dv 

—-—  =  —  r—  for  any  point  "n",  hence  if  w  is  a  con- 
dt   wn  u t 

tinuous  function  of  s,  we  have 
dv   1     dvt,     1 


wt  -   hence  the  line  integral 
dt   w      dt  dt   of  the  acceleration 

along  a  stream  lines,  becomes, 


299 


dv     d_v  ds 
3  dt  "  "ldt  f(s) 

The  line  integral  of  the  acceleration  may  be 
obtained  to  a  sufficient  degree  of  exactness  by 
dividing  stream  lines  into  a  linear  group  of  columns 
of  various  sections,  obtaining  the  proper  acceleration. 
To  form  (8)  and  multiplying  by  the  length  of  the  res- 

pective columns  and  then  adding  these  columns  together. 

1        dv 

The  term  -  /  ds  (—  —  )  is  found  usually  to  be  relatively 
g        dt 

small  compared  with  the  pressure  drops  due  to  throttling 
and  the  changes  of  pressure  due  to  changes  of  section. 

Hence  (7)  reduces  to  the  energy  equation  for 
uniform  or  steady  flow,  known  as  Beraoulliis  theorem, 
that  is, 

t  t 


P        T 


D      2    D   z   2g 
p       v* 

The  term  —  *•  +Zt+  —  x  is  known  as  the  total  head  at 

section  (1),  composed  res- 

pectively of  a  pressure  head,  gravitational  head  and 
a  velocity  head, 

(4)   When  friction,  viscosity  or  turbulent 
motion  occurs  Bernoulliis  theorem  is  modified  by  a 
friction  head  hf. 

Considering  a  tube  of  a  stream,  we  have  for  steady 
motion  Dw  ds 

Pt*td«^t!ftd«,+D"t<Ut(*t-Zt)-ar   =   *   *  (v*-v*)  (10) 

2     2g 

Where  d  Wf  corresponds  to  the  differential  work 
due  to  friction  for  a  differential  quantity  of  flow 
d  ft. 

By  the  equation  of  continuity,  dQ  =  wtdsi=  ",dsf 

hence  (9)  reduces  to 

Pt-pt+D(zt-zt)-^»-(vX>         (ID 

dWf  i 

but  -  •>-  =  hf  i   ,  known  as  the  head  loss  due  to 
dft       2 

friction  between  1  and  2,  hence 


300 


t  i 

<r  *  5  *  z.>-  (r  *  if  *  z.)  •  h<; 

It  is  to  be  especially  noted  that  in  the  flow  of  a 
liquid  through  an  orifice  as  in  a  recoil  brake,  the 
•ajor  loss  is  in  the  nature  of  a  frictional  loss  of 
head  due  to  the  contraction  and  sudden  expansion  of 
the  liquid  through  the  orifice,  thus 


u 

"  *   = 


in  equation    (3),    that    is 


p,-p.  r  "' 

—  -  -  =  -  7      since  Z  =  Z  approximately,  also 
*a 


t         x 

Vt  V2 

—   and  -   are  relatively  small. 
2g        2g 

It  may  be  shown  by  a  somewhat  similar  analysis 
that  in  the  consideration  of  friction  of  or  turbulent 

loss  of  head  due  to  throttling,  that,  from  (7)  we 
have 

2  8 


where  if  wo  =  the  area  of  the  jet 

VQ  =  the  velocity  of  the  jet. 
where  hfi   has  the  form, 


Hence  anaxact  expression  for  a  stream  line 
passing  through  a  jet,  and  the  whole  stream  line 
itself  under  acceleration,  becomes, 


D    2g        D  2g      g  dt     w0 

(5)     The  pressure  variation  across  a  stream 
line  may  be  obtained  by  a  consideration  of  a  cylinder 


301 


the  end  faces  of  which  are  in  the  outer  and  inner 
boundary  surface  of  a  stream  line  tube,  and  the  axle 

.is  perpendicular  to  the  stream  line  axis.   We  have, 
if  w=  cross  section  of  differential  cylinder 
r  =  radius  of  curvature  of  stream  line 

d  =  height  of  differential  cylinder 
0  =  angle  between  r  and  the  vertical 

that  a 

Dw          v 
(p+dp)w-pw+Dwdr   cos  0  =  —  y  dr  —  (15) 

2 
V 

wdp   =   Dwdr(  --  cos  0) 
gr 

j  g 

hence  -7*-  =  D  (-  --  cos0)  (16) 

dr       gr 

which  given  the  rate  of  change  of  the  pressure  across 
a  stream  line  with  respect  to  the  radius  of  curvature. 
Neglecting  the  weight  component,  we  have, 

£  -  — 

dt    gr 

Hence  for  circular  or  vortex  motion,  the  change  in 
pressure  along  the  radius,  becomes, 


dr 


In  particular  if  the  total  system  acquires  the 

sane  angular  velocity1"'-  ,  we  have, 
—  dt 

1_  =  r  (*i)« 

r       dt 


since  the  total  head  at  any  point  in  a  fluid  equals, 

2 

p        v 

H  =  ~  +  Z  +  r—   the  variation  of  head  across 
D        2g 

dp     vdv 
a  stream  line  becomes,   dH.  -  ——  +dZ+  - 

L   *  vdv  \  (2o) 

=  r*-  +dr  COS0+  -   ) 

D  g 

Substituting  Eq.(16)  in  (29)  we  "have, 


302 


_•      vdv 

dH=  —  dr  +  (21) 

8*       6 

which  is  the  general  equation  for  the  change  in  head 
across  a  stream  line. 

(6)  When  the  flow  is  radial,  evidently  the 
flow  outward  fro»  circumferences  of  various  radii, 
becones,  Q  »  2n  rv  =  2*  rv  ,  hence  voro»vr 

or  _ 

v  ro 

—  =  —   and  for  steady  motion,  we  have, 

v_  r 
o  x  2 

!°.  +  Is  +  -  L  +  L.  ,  z 

D    2g    0=  D  +  2g 

hence       *     * 

°/,   *i  /««x 

P-P0=  2ja~  7»)  (22) 

In  terms  of  the  total  head  H,  we  have, 

*     2 

P       vo  ro 

r  • H  -  -3^-        (23) 

(7)  A  free  circular  vortex  occurs  when  the 
total  head  of  any  annular  stream  line  of  the  vortex 
is  the  same. 

That  is,  for  any  annular  stream  line, 

t 

p    v 
fl  =  -jj—  +  — —  =  const. 

To  find  the  distribution  of  pressure,  we  have, 

dp   vdv 

~~  "*       °   and  for  the  flow  slowly  out- 
u    g 

ward  radially,  we  have, 

P-Po   vo"v* 

— - —  «     •»-    (Neglecting  friction) 
D      2g 


Thus  the  pressure  variation  is  exactly  similar 
to  that  of  ordinary  radial  flow. 
How  from  (21)  since  dH  »  0 


303 


v       vdv  dv     dr 

—  dr  +  =  0     hence  —  = and  from  the  la* 

gr       g  v      r 

of  continuity 

for  the  flow  outward,  v  r  »  v  r  hence  v  -  JL 

r 

Likewise  the  flow  outward  is  exactly  similar 
to  ordinary  radial  flow. 

THE  EFFECT  OF  THE       The  viscosity  of  a  fluid  is 

VISCOSITY  OF  FLUIDS.   the  shear  stress  to  the  distortion 

of  the  fluid  and  this  stress  is 
measured  by  the  coefficient  of 
viscosity  times  the  rate  of 

distortion.   In  other  words  the  viscosity  or  coefficient 

of  viscosity,  becomes, 

s  dv 

u  =  —      or  s  =  u  jr    where  v  *  velocity  of  a 

—  lamina  flow  (ft/sec) 

h  =  normal  to  flow 

lamina   (ft) 
s  3  shear 
1°  Flow  between  flat  surfaces: 


r 


dv 

S  =   u  —   .bl 
dh 

d2v 

dS*u  rrr  dh.bl 
dh' 

Now  considering  the-  forces  on  a  lamina  of  thick- 
ness dh,  breadth  b  and  length  1,  we  have,  for  a  constant 
pressure  head  (pt~p  ) 

(p  ~P  )  bdh-dS  =  0   for  uniform  flow 


304 


(p  -p  )bdh-bln  ^-7  dh  *  0 
dh 


d'v 


(pt-Pt>-lu  —7 

dv 

—  » 


dv    (Pt-P. 

Integrating,  us  have  —  »       '    +  C 


dv 
when  —  =  0,    Ct  =  0 

(p.-pX 

Integrating  again,   v  =  —  -  —  -  -  *  C.  when  v  *  0, 

2ul 


(Pl-P,)Hf 
and  Cg  =  -  -     Hence  the  distribution 

of  velocity  across  a 
section  is  given  by  the  equation, 

<Pt-Pt>   *   ^ 

v  =   ^ul  -  (^  ""  7"^  (ft/sec)  as  measured  from  the 

center.   For  a  dif- 
ferential flow,  we  have 

P  -P,    2   H* 
dQ  =  vbdh  =  ~  —  -  (h  --  )  bdh  and  for  the 

2ul  total  flow, 

summing  up  oo  both  sides  of  center  line,  we  have, 


Q  =  —  -  bH    Therefore  the  drop  of  pres- 
12   ul 

sure  between  flat  surfaces 

in  a  rectangular  channel  becomes. 

12ul   A     ,.,  N 
pt-p2  =  -  ft     (Ibs) 
bH» 

For  the  particular  case  of  a  square  section, 


12ulQ 

Pi-Pa  =  "TT"        (Ibs) 
n 


305 


2°  Plow  through  a  circular  section: 


p,  1 

p 

1  1 

1 

1 

1      2- 
I         rh 

1 

<L 
i 

dv 

The  viscosity  shear  becomes,  S  =  -  2nrlu  — 

dr 
(r  ID 

dr 

dS  =  -  2nuld  — dr 

dr 

Considering  the  forces  on  a  cylindrical  lamina  of 
thickness  dr  and  length.  1,  we  have, 
(pt-p2  =  2iirdr  -  dS1  =  0  for  uniform  flow 

AI   d^ 
T— ) 

(pt-p,)r*ul  dr  r   =  0 

Integrating,  we  have        ^p  _p  )r* 

r  —  =  -  — - — +  C 

dr      2ul 

which  may  be  written 


dr      2ul       r 

dv  C 

when  —  =0,  r  =  0,  hence  —  *  0  i.e.  C  =0  since 
dr  r 

r  may  have  any  finite  value.  Integrating  again 

v  =  -  — — — —  *  C-2  when  v  =  0  for  the  boundary 
4ul 

surface, 

r  »  R 


306 


t, 
and  v  =  -i-i  (R*-rf) 


which  gives  the  variation  of  the  velocity  over  a 

cross  section  as  a  function  of  the  radius  from 
the  center. 

For  the  total  flow  per  second,  we  have 

r 

Q  »  /   2nrdr.v 
o 


/  (R'-r')rdr 


2  ul    o 

(pt-Pt)nR4 

8  ul 

Hence  for  the  drop  of  pressure  through  a  small  orifice 
where  there  is  no  abrupt  change  in  section, 

8  ul 
Pi~pa  =  —  4~"  Q   where  u  =  coefficient  of  viscosity. 

71  R 


that  is  the  drop  of  pressure  varies  as  the  length  and 
inversely  as  the  4th  power  of  the  diameter  of  the 
orifice. 

PRINCIPLE  OF  MOMENTUM      The  various  formulae 
AND  DYNAMIC  REACTIONS.   previously  developed  de- 
pended upon  the  ap- 
plication of  Bernoulli's 
theorem,  or  the  energy  equation 

of  hydro  dynamics.   A  theorem  of  equal  importance  is 
the  principle  of  momentum  and  from  it  with  a  com- 
bination of  Bernoulli's  theorem,  we  may  compute  the 
various  dynamic  reactions,  that  occur  in  hydro 
dynamic  problems. 

If  P  =  the  unbalanced  reaction  on  a  mass  of 
water  and  the  velocities  of  the  mass  is  changed  in 


307 


time  t,  from  vt  to  V2  ft/sec,  then  Pt=ra(v  -v  ) .   In 
the  application  of  this  principle 

(1)  the  mutual  reactions  between  the 
particles  of  water  are  obviously  en- 
tirely neglected. 

(2)  the  velocity  components  in  the 
direction  of  motion  are  only  to  be 
considered. 


DERIVATION  OF 

THROTTLING 

FORMULAS. 


(1) 


Loss  of  Head  due  to 


sudden  expansion. 
In  the  flow  of  a  fluid  through 
an  orifice  the  drop  in  pres- 
sure or  loss  of  head,  is 

primarily  due  to  the  sudden  expansion  or  abrupt 
change  from  a  small  to  a  large  section,  of  the  fluid 
flow.   The  loss  of  head  is  due  to  the  formation  of 
eddies  due  to  the  sudden  expansion  of  the  flow  and 
the  consequent  dissipation  of  energy.   If  the  cross 
section  of  the  flow  is  gradually  enlarged  from  that 
of  a  small  orifice  to  a  large  section,  no  eddies  are 
produced  and  we  have  no  loss  of  head.   Thus  in  the 
Venturi  meter  the  fluid  passes  from  a  large  section 
to  a  very  small  section  and  then  back  again  to  a 
large  section  but  since  the  change  in  section  is 
gradual,  we  have  no  drop  in  pressure.   Hence  we  have 
a  very  important  principle  that  is  fundamental  in 
the  design  of  recoil  throttling  orifices:- 

A  throttling  drop  in  pressure  cannot  be  pro- 

duced  without  a  sudden  change  in  section  of 
the  flow  of  the  fluid. 


Y*\//t///jfa//l 


308 


Consider  a  flow  of  fuild  passing  through 

sections,  ab,  mn  and  cd  respectively.   Let  the 
cross  sections  of  the  stream  be  w   at  ab   w  at  cd 

1  12 

and  the  corresponding  pressures  be  pt  and  p  res- 
pectively. Let  pQ  be  the  pressure  at  mnt  (Ibs/sq.ft). 

From  the  energy  equation,  we  have, 

t         * 

P!     Vl     P2      Vt 

—  +  —  =  —  +  —  +  hj     assuming  a  continuous 
D    2g   D    2g          uniform  flow. 

From  the  principle  of  momentum,  we  have 

• 

Q 

Piwi*Po(w2~wi)~P2W2=  g~  D  (v2~vi)     where   Q  =   the  rate 

of  flow  (cu.  ft/sec) 

D  =  the  density  of  fluid  (Ibs/cu.ft).   Now  from  the 
experiment  it  is  found  that  po  =  pt  hence  the  momentum 
equation  reduces  to, 

ft  _,  Pi     P2     T2(V2-Vl) 

(P-P"   D(v-v)  and—   —  = 


at 
hence 


g 
(v-v)2 


which  simplifies  to 


tt 

hf=  -  or  in  terms  of  the  area  of  the  orifice 
2g 

and  the  enlarged  section,  since  wtvt  =  *2v2 

«'  2' 

T        W        V    W 

hf  =  -i-  (1-  -i)2  =  -i  (-1  -  1) 
2*     w2     2g  wt 

(2)   Loss  of  Head  due  to  sudden  contraction, 
When  the  cross  section  is  suddenly  diminished 
beyond  the  reduced  section  we  have  eddying  of  the 
flow  with  a  resultant  loss  of  head.   This  too  is 
really  a  special  case  of  (1)  since  just  beyond  the 
contracted  section,  the  stream  becomes  even  more 
contracted,  followed  by  a  sudden  expansion  until 
the  stream  reaches  the  cross  section  of  the  con- 
tracted area. 


309 


Therefore  if  w  =  the  area  of  the  contracted  section, 
then  the  cross  section  of  the  contracted  stream  be- 
comes, c  w  where  c  =  depends  upon  the  preceeding 
area  w  and  the  area  of  the  orifice.   In  terms  of  the 
velocity  of  the  orifice,  the  loss  head  equals 

1  2 

V      W  V 

hf=  — —  ( -  1)   =  E-r —  where  if  w  =  the  cross 

2g   ex  2g 

section  before  the  sudden 

contraction  of  section,  the  experinents  of  Weisbach 
give  for 


o.i  o. 624  o. 360 

0.2  0.632  0.340 

0.30  0.643  0.310 

0.40  0.659  0.266 

0.5O  0.681  O.220 

O.6O  O.V12  0.162 

o.7o  o.'/BS  0.106 

O.8O  0.813  0.053 

0.90  0.892  0.014 

l.OO  l.OO  O. 


IB  the  special  case  when  — —  =  0,  that  is  the  area  of 
the  orifice  w±  is  entirely  negligible  with  the  flow 
from  the  large  cylinder  as  in  a  flow  from  a  resevoir, 
c  =  0.6  and  E  =  0.445. 

It  is  to  be  noted  that  the  above  analysis  holds 
only  when  the  length  of  orifice  is  sufficiently  long 
to  allow  the  contracted  stream  in  the  orifice  to  ex- 
pand and  completely  fill  the  orifice  before  expanding 
in  the  region  beyond  the  orifice.   Hence  the  loss  of 
head  due  to  sudden  contraction  only  holds  for  long 
orifices  or  entrances  into  long  channel  parts. 


310 


LOSS  OP  HEAD  AND  PRESSURE       Assuming  uniform  flow 
DROP  THROUGH  RECOIL  ORIFICE.    from  the  recoil  cylinder 

of  effective  area  w, 
through  an  orifice  of 
cross  section  wt  dis- 
charging into  a  cylinder  or  channel  of  cross  section 
w  .   Then  from  section  w  to  the  mid  section  of  the 
orifice, 

v*    P    v2 

5  +  jj-  -  -5  +  jj  +  hfc  (1)   hfc  =  loss  of  head 

due  to  sudden 
contraction 

and  from  the  mid  section  of  the  orifice  w  to  the 
rear  of  cylinder  or  channel  cross  section  vr  , 

2  a 

^  *  5?  =  ^D  +  55  +  hfe  (2)   hf e  =  loss  of  head 

due  to  sudden 
expansion. 
Adding  (1)  and  (2),  we  have 

P    /    P2    v* 

—  +  —  =  —  +  —  -«•  hfc  +  "f  e 

D    2g    D   2? 

Very  often  v  =  va  approximately  and  usually  the  heads 
corresponding  to  v   and  v   are  small  compared  with 
the  pressure  and  throttling  heads  and  therefore  the 
velocity  heads  may  be  entirely  neglected.   We  have 
then, 

P  -  P. 
1—  =  hfc  +  hfe     that  is  the  drop  in 

pressure  through  an  orifice 
is  equal  to  the 

total  head  lost  due  to  sudden  contraction  and  ex- 
pansion. 

Now 

?'  2' 

V   W          .V       VI    . 

hf  -  -i(-±  -  D'=  ~U-  -1)2 
2g  w         Xg    w2 


311 


*f*  *  *T* 


further  AV  =  w^*  wfvf, (where  V-  the 
velocity  of  the  recoil  piston.) 
(w=A  =  effective  area  of  recoil  piston) 
hence 

A*Vf<      w 

^r(1-^ 


When  the  orifice  is  in  grooves  in  the  cylinder  or  through 
orifices  in  the  piston,  we  have 

)since  *t=A  approx- 
iraately, 


v  =V  approximately  and 


"^  *  c7"  aPProximate1^ 


( 

)the  effective  area 
(of  the  recoil  pis- 
)ton  and  wt=effective 


j^ 

Therefore  if  n  -  —  ,we  have 


V   .   ,.t' 
—  (m-1) 


(area  of  orifice, 

)that  is  the  contracted 

(flow  through  the 

)orif ice. 

(c  =  coefficient  of 

)contraction  of 

"* 

(orifice. 

Usually  the  loss  of  head  due  to  sudden  contraction 
nay  be  entirely  neglected  as  compared  with  the  loss 
of  head  due  to  sudden  expansion,  hence  we  have,  for 
a  very  close  approximation  of  the  pressure  drop, 
(1)     With  throttling  through  grooves 
in  cylinder  or  piston, 

p-p     V8 
2 (m-i)» 

D     2g 


or 


.*'«*' 
A  V 


cw 


both  forms  having  useful  applications. 


of 
contraction, 


312 


(2)     With  throttling  through  an  orifice 
froa  the  recoil  cylinder  to  the  re- 
cuperator, 


»i  «  area  of  orifice 

w  -  area  of  channel  leading  from  orifice. 

ANALYSIS  OF  THE  MUTUAL        In  an  ordinary  brake 
REACTION  IN  A  RECOIL  BRAKE.   cylinder  we  have  a  groove 

in  the  cylinder  or  at  the 
circumference  of  the  re- 
coil piston. 

As  the  recoil  rod  pulls  out,  a  pressure  is  created  on 
the  front  side  of  the  piston,  due  to  the  forcing  of 
the  fluid  through  the  orifice  groove.    The  pres- 
sure is  by  Berboull's  theoren,  obviously  lowered  in 
the  vicinity  of  the  orifice  due  to  the  increased 
velocity  of  the  flow. 

Hence,  with  an  orifice  in  the  piston,  the 
sure  is  not  uniformly  distributed  over  the 


effective  area  of  the  piston. 

Therefore,  the  brake  reaction  is  not  equal  to  the 
product  of  the  recoil  cylinder  and  the  effective  area 
of  the  recoil  piston. 

Let  X  =  the  total  reaction  of  the  fluid  on  the  re- 
coil piston.  (Ibs) 

p  *  the  pressure  in  the  recoil  cylinder.  (Ibs/sq. 

ft) 
A  «  effective  area  of  recoil  piston. 

=  0.7854  (D»  -  d£  )   (sq.ft) 
where  Dr  =  dian.  of  recoil  cylinder  (ft) 

dr  »  diam.  of  recoil  rod   (ft) 

Ar  *  area  of  recoil  brake  cylinder  =  0.786  Isq.ft) 
V  »  velocity  of  the  recoil  piston  (ft/sec) 
v  *  velocity  of  flow  through  the  orifice,  (ft/sec) 
D  »  density  of  fluid   (Ibs/cu.ft) 
w  »  area  of  orifice   (sq.ft) 


313 


c  =  contraction  factor  of  orifice. 

Assume  the  recoil  rod  to  only  extend  from  one  end 
of  the  piston.    In  this  case,  we  have  a  void  in  the 
rear  of  the  piston  due  to  the  volume  displacement  in 
the  front  of  the  piston  being  less  than  in  the  rear 
of  the  piston. 

Assuming  the  pressure  in  the  orifice  to  be  small, 
we  have,  for  the  reactions  on  the  fluid  from  front 
head  of  cylinder  to  a  cross  section  at  the  center 
of   the  orifice:- 


(1) 


and  the  reactions  on  the  fluid  from  the  orifice  to 
the  rear  head  of  cylinder,  becomes, 


The  reaction  on  the  piston  =  X  to  the  rear 

The  reaction  on  the  cylinder  =  pA  -  Y  to  the  front 

Adding  (1)  and  (2)  we  have,  pA>  -  X-Y  =  0,  which 
is  immediately  obtained  since  there  is  no  change  in 
the  total  momentum  of  the  fluid,  as  we  should  expect 
from  first  principles,  since  the  fluid  acts  as  a  medium 
for  the  transmission  of  the  reaction  between  the  re- 
coil cylinder  and  the  recoil  piston.   Hence  pA  -  Y  =  X. 

which  gives  the  actual 
reaction  exerted  on  the 

"brake  piston.   Since  C  vr  v  =  A  V,  by  the  law  of  con- 

tinuity, then 

£    2 

AV          Cwv     ,  v        D  A  v    ,    . 
v  -  —  and  V  =  -  and  X  =  pA  -  -   (ibs) 
cw 


Dv* 
now  p  =  ,  from  Bernoulli's  theorem, 


314 


Hence  X  =  —  —  (A  -  2  cv») 
2g 

=  °  \  I   (A-2  cw)  (Ibs) 
2gc  w 

D  v*   DA*  Y2 
but  p  =  -  =  -  hence  X  =  p(A-2  cw) 

2g     2gcawa 

That,  is  the  reaction  on  the  piston  equals  the 
product  of  the  pressure  in  the  recoil  cylinder  and 
the  effective  area  of  the  recoil  piston,  where  the 
effective  area  of  the  recoil  piston  equals  the  an- 
nular area  betneen  the  recoil  cylinder  and  piston 
rod  decreased  by  twice  the  contracted  area  of  the 
orifice. 

A  physical  explanation  i$  that  due  to  the 
pressure  of  the  orifice,  we  have  the  pressure  lowered 
around  the  orifice.   Hence  we  must  not  only  subtract 
the  area  of  the  orifice,  but  also  an  additional 
equivalent  area  which  is  to  account  for  the  lowered 
pressure  about  the  orifice. 

Since  c  =  0.6  approx.,  then  2  cw  =  w  approx., 
and  therefore  for  practical  calculations,  the  an- 
nular area  of  the  recoil  piston  is  merely  decreased 
by  the  total  throttling  area  through  the  piston. 

Ifhen  the  rod  is  assumed  to  extend  through  both 
ends  of  the  recoil  cylinder,  we  have  a  continuous 
rod  in  the  cylinder  and  therefore  no  void  is  pro- 
duced during  the  recoil. 

Assuming  the  same  symbols  as  before,  we  have, 
since  the  total  change  of  momentum  of  the  fluid 
is  nil,  pA  -  X  -  Y  =  0.   Hence  X  =  pA  -  Y  and  the 
fluid  merely  transmits  the  mutual  reactions  be- 
tween the  recoil  cylinder  and  recoil  piston. 

Let  pw  =  the  pressure  in  the  orifice.  (Ibs/sq.ft) 


)  Xf  =  total  reaction 
(      on  front  of  re- 
)      coil  piston. 
for  the  momentum  of  the  fluid  contained  in  the  front 


315 


part  of  the  cylinder  to  the  orifice,  and 

DAY 
Y-pww-Xr=  -  v    )  Xf  =  total  reaction  on 

(      rear  of  recoil 

)      piston 

for  the  momentum  of  the  fluid  contained  from  the 
orifice  to  the  rear  end  of  the  cylinder.   Now 
X  =  Xf-Xr=  total  reaction  on  recoil  piston.   Due 
to  the  sudden  expansion  of  the  fluid  after  leaving 
the  orifice,  the  pressure  on  the  rear  face  of  the 
piston,  becomes,  pw(A-w)=Xr  (assumption  from  ex- 
periment -  sudden  expansion),  hence 

Y  -pwA  =  D  A  V  v   and  X.  =  Xf-pw(A-w) 


DAY 
=(p-pw)A  --  -  —  v 

Dv* 
Applying  Bernoullis'  theorem,  we  have  p-pw  =  - 


but  by  the  law  of 
continuity  c  w  v  =  A  V  therefore 


Dv* 

=  — —  (A-2-cw)   (Ibs)  Since  pw  is  negligible 
*f 

compared  with  p,  we  have 


Dv2 
P  -  Pw  -  P  = 


2g 

hence,  as  before  X  =  p(A-2cw) 

That  is  the  total  reaction  on  the  recoil  pis- 
ton equals  the  product  of  the  pressure  in  the  re- 
coil cylinder  and  the  effective  area  of  the  re- 
coil piston,  when  the  effective  area  of  the  recoil 
piston  equals  the  annular  area  between  the  recoil 
cylinder  and  piston  rod  decreased  by  twice  the  contracted 


316 


area  of  the  orifice. 

Since  c  =  0.6,  2cw  =  w  approx.,  and  therefore 
again  for  practical  calculations,  the  annular  area 
of  the  recoil  piston  is  merely  decreased  by  the  total 
throttling  area  through  the  recoil  piston. 

DERIVATION  OF  RECOIL     We  may  consider  the  throttling 

THROTTLING  FORMULAS,   effected  in  either  of  the  follow- 
ing manners:  (1)  throttling 
through  grooves  in  the  cylinder 
wall  or  through  a  variable 

orifice  in  the  piston  itself, -(2)  throttling  through 

a  stationary  orifice. 

(1)     Throttling  through  a  variable 

orifice  in  the  piston  or  grooves  in 


the  cylinder  walls. 

Let 


A  =  effective  area  of  the  piston,  i.  e.  the 
cross  section  of  the  cylinder  minus  the 
cross  section  of  the  rod.    (sq.ft) 
p  =  the  intensity  of  pressure  at  the  pressure 

end  of  the  cylinder  (Ibs/sq.ft) 
D  =  the  density  of  the  liquid   (Ibs/cu.ft) 
V  =  the  velocity  of  the  recoil   (ft/sec) 
w  =  the  area  of  the  orifice     (sq.ft) 
v  -  the  velocity  of  flow  through  the  orifice. 

(ft/sec) 
X  =  the  total  fluid  reaction  against  the 

piston  (Ibs) 

Then,  we  have, (neglecting  the  small  pressure  in  the 
orifice)       D  A  v 

pA  -X  »  v  -  -  -  for  the  momentum 

generated  in  the  jet, 
Dvz 
and  p  = _____  for  the  energy  of  the  flow 

in  the  jet. 
AV  =  cwv  -----  from  the  law  of  continuity  of  the 

flow, 
then 


317 


DA     2cw  2 
X  =  ~  (1-  — ), 

3   2 

DA  V     ,    2cw, 

(1 r-)     (Ibs) 


2gc*w2       A 

Since  the  reaction  on  the  cylinder  is  the 

difference  between  the  force  pA  at  the  pressure  end 
and  the  reaction  of  the  jet 

D  A  V 

-  v  flowing  from  the  orifice  we  have  the  reaction 

on  the  cylinder  also  equal  to 
D  A  V      DA3V*      2cw 


as  would  be  expected  from  the  equality  of  action  and 
reaction. 

Ulith  a  continuous  piston  rod  through  both  ends  of 
the  cylinder  we  may  neglect  the  pressure  through  the 
orifi*ce  and  since  by  experiment  the  pressure  on  the 
rear  face  of  the  piston  is  practically  that  through 
the  orifice,  the  reaction  on  the  piston  remains  the 
same.  Here  again  the  reaction  on  the  cylinder  is 

DAY  ..   D  A  V 

pA-p  A  =  pA  --  v,   since  p  A  -  -  v   as  would 

5  3 

be  expected  from  tlie  equality  of  action  and  reaction. 
The  reaction  X  on  the  cylinder  may  be  written 

Y   PA^2        2cw 
X  = 


2gc*wa        A  ' 

Dv2 

Further  since  p  =  =  — • — ,  ,   we  have  also. 

Zg    2gcaw* 

X  =  p(A  -  2  cw) 

=  p(A  -  w)  approximately. 

Thus,  knowing  the  pressure  in  the  pressure  end 
of  the  recoil  cylinder  to  obtain  the  reaction  on  the 
piston,  we  must  multiply  this  pressure  by  the  ef- 
fective area  of  the  piston  minus  the  area  of  the  re- 
coil orifice. 

(2)     Throttling  through  a  stationary 

orifice. 
With  a  stationary  orifice,  the  throttling 


318 


usually  takes  place  between  the  recoil  or  brake 
and  recuperator  cylinders.   The  loss  of  head  or  pres- 
sure drop  is  mainly  due  to  the  sudden  expansion  of  the 
flow  from  the  orifice,  though  with  a  relatively  long 
orifice  the  loss  due  to  sudden  contraction  may  become 
appreciable. 
If 

w  =  the  area  of  the  orifice  (sq.ft) 
A  =  the  effective  area  of  the  recoil  piston 

(sg.ft) 

V  =  the  velocity  of  recoil  (ft/sec) 
v  =  the  velocity  through  the  orifice   (ft/sec) 
c  =  contraction  factor  of  the  orifice. 
H  =  the  area  of  the  channel  leading  away  from  the 

orifice,  (sq.ft) 
Then  from  Bernoulli's  theorem,  we  have 

p-pa  )  where  p  =  the  pressure  in  the 

~~~ =  ^t        (          recoil  cylinder. 

)  pa  =  the  pressure  in  the 
(          recuperator. 

Mow  )  hf  =  total  head  lost  due  to 

hf=hfc+h.fe     (          throttling. 

)  ^fc=  l°ss  °f  head  due  to 
(  contraction. 

)  nfe~  l°ss  °f  head  due  to  ex- 
(          pans  ion. 

T* 

Now  hf.,  =  £  —  where  5  may  be  taken  0.35  to  0.5  and 
gf 

and       *  z 

v   ,    cw.a  v  ...   cw.*   f. 

hfe=  "^  (1  ~  "1L)    hence  hf=  ~~[(1 >   *  *] 

2«  2g      W 

In  recoil  mechanisms  W  is  usually  made  from  2.3  to  3.0 
tines  w.  Then,  we  have,  if  c  is  taken  approximately  = 
0.65 

(1  --J.)*  =  0.515  to  0.614 
For  flow  from  an  orifice  into  a  large  reservoir 


319 


4  >  0  and  (1  -  £-)*  <  1 

n   * 

Hence  usually 

cw  « 
[(1  -  -T-  )   +  &]  =  1  approximately, 


D  A*V» 

-55   hence  p-p.  =  -      for  the  drop  of 
2gcawz 

pressure  through 

the  orifice.  The  reaction  on  the  recoil  piston  is, 


D 

X  =  pA  =  -  —  .  .   +  p.A 
** 


In  recoil  design,  it  is  customary  to  measure 
areas  in  sq.  inches  and  pressures  in  Ibs/sq.in. 
Further  the  average  specific  gravity  of  the  re- 
coil oils  used  in  our  service  may  be  taken  at  0.849 
and  therefore  the  density  D  becomes,  D  =  62.5  x 
0.849   (Ibs/cu.ft). 

The  recoil  throttling  formulas  become,  therefore 
(1)     For  throttling  through  a  variable 

orifice  in  the  piston  or  grooves  in 


the   cylinder  vralls:- 

X   = 
P  = 

6  K2A»V2 

(Ibs)            w  = 
(Ibs/sq.in); 

KA*V       /6~ 

(sq.in) 
(sq.in) 

175  w2 

13.2    /x 
KAV 

175   w* 

where  K  = —  =  1.6  to  1.3  approx. 

6=1 •—:        c  =-  0.6  to  0.8  approx. 

(2)     For  throttling  through  stationary 


X  = 


orif ices:- 

KAV 


175 


320 


~P*  *  175«*         (Ibs/sq.in) 


CW    ft 

where   K  =   1.6  to   1.3   approx.    6  =(1  --  )      +   E 


VARIATION  OF  THE  THROTTLING  ffe   have   seen   the 

CONSTANT   IN  THE   RECOIL  total   braking  on   the 

recoil  piston  may  be 
expressed,    when 
throttling   through   a 

variable  orifice    in  the  piston   or   through  grooves 

in  the   cylinder,    as 

K«  A»  V« 


and   when   throttling  through  a   stationary  orifice, 
as 

XIL  I     IV     A     V  f  •*  *        \ 
=    a     -  +  p.A       (Ibs) 
175w« 

.                 2cw  .i                   cw.a        _ 

where   6=1  --  -  and  o     =   (1  --  )     +  £ 


Since  w  varies  throughout  the  recoil,  6  and  6'  must 
also  necessarily  vary  in  the  recoil.   Calculations 
with  the  omission  of  the  term  6  or  6*  have  been 
found  slightly  in  error  and  this  error  has  been 
ascribed  to  variations  in  the  contraction  factor 
of  the  orifice.   The  contraction  factor  may  also 
vary  but  it  seems  more  probable  that  the  error  is  due 
to  the  omission  of  the  term  6  or  61  . 

With  stationary  orifices  -^  and  5  can  very- 

H 

often  be   neglected  and   therefore  the   variation   in 
the   throttling  constant  can  be  neglected.      With 
throttling    through   the  piston   or  by  grooves   in 
the  cylinders  -2filL    is   small   but   not   negligible^ 

hence   with   this   type   of   throttling  variations    in 
the   orifice   are   more  marked. 


321 


For  a  preliminary  design  6  and  6'  may  be 
assumed  equal  to  unity;  but  on  recoil  analysis  and 
careful  tests  6  and  its  variation  in  the  recoil 
should  be  taken  into  consideration. 


CHAPTER   VI 
DYNAMICS  OP  RECOIL. 


ELEMENTARY  PRINCIPLES.     The  object  of  the  recoil 

is  to  reduce  greatly  the 
stresses  induced  in  the  car- 
riage.  Without  recoil,  the 
reactions  brought  on  the 

various  parts  of  the  carriage  are  direct  functions 
of  the  maximum  powder  force,  which  would  require  a 
very  massive  carriage  for  guns  of  large  caliber. 
The  mutual  reactions  created  by  the  powder 
gases  between  the  gun  and  the  projectile  is  of 
very  short  duration  compared  with  the  time  of  recoil 
and  for  a  rough  approximation  nay  be  treated  as  an 
impulsive  reaction.   Neglecting  the  mass  of  the  pow- 
der gases,  we  have  /Pdt  =  mv  and  /Pdt  =  MV.  Therefore 
mv  =  MV,  where  m  =  mass  of  the  projectile 

M  =  mass  of  the  recoiling  parts 
v  =  velocity  of  projectile 
V  =  velocity  of  recoil 
/Pdt  =  impulsive  reaction  of  the  powder 

gases. 

The  momentum  generated  by  the  action  of  the  pow- 
der gases  in  the  projectile  and  gun  is  the  same,  as 
is  immediately  obvious  from  the  principle  of  con- 
servation of  momentum.   It  is  to  "be  further  noted 
that  finite  forces,  as  the  resistance  to  recoil,  can 
be  neglected  in  the  consideration  of  impulsive  actions, 
and  since  the  generated  velocity  of  recoil  acts  for  a 
differential  time,  the  recoil  displacement  during  the 
impulsive  action  can  also  "be  neglected. 

The  kinetic  energy  of  the  recoiling  parts,  after 
the  impulsive  action, is 

A.  ~±  MV' 
Since  V  = ,  the  recoil  energy  in  terms  of  the 

323 


324 


IB 

velocity  of  the  projectile  becomes,  A  =  —  (-  mv  ). 
Hence  the  energy  of  recoil  is  but 

n 

-  of  the  energy  of  the  projectile. 

The  total  energy  generated  by  the  impulsive 
action  of  the  powder  gases,  is,  therefore 

i      m 

-  (i  .  5>« 

Obviously  the  greater  M,  the  smaller  the  energy 
of  recoil. 

The  reaction  R  between  the  gun  and  raount  for  a 
recoil  displacement  b,  is     -  MVa 

R  =  ^ 

or  in  teras  of  the  velocity  of  the  projectile 
_ 

"*    ,  *     *  \ 

~H  (;  BV  } 

The  reaction  is  thereby  reduced  proportionally 
to  the  increase  of  ths  recoiling  mass  M.   Hence  to 
reduce  the  recoil  reaction  we  increase  the  recoiling 
mass  14  and  the  length  of  recoil  h  . 

The  dynamical  relations  for  an  elementary  recoil 
analysis  in  terns  of  the  relative  velocity  of  the 
projectile  with  respect  to  the  gun  vp  can  "be  readilj 
obtained  as  follows:- 

Vp  =  v  +  V  assuming  V  measured  in  the  direction 
of  recoil  from  the  conservation  of 
momentum         m  Vg 

MV  *  mv   =   m(vR-  V):    hence   V  =  - 

M  +  m 

The  energy  of   recoil   is 


and  the  recoil  reaction 


If   the   recoiling  parts   are  hrought   to   rest  hy 
friction    alone,    R  =    u   Mg 


325 


1  V2 
hence  b  =  -  — •  3 

2  ug 

DOUBLE  RECOIL  SYSTEM: 

When  a  gun  is  mounted  on  a  movable  mount  as  a  car 
body  or  itself  rolls  along  a  plane,  we  have  virtually 
a  doubl.e  recoil  systen,  the  upper  recoil  being  between 
the  gun  and  mount,  and  the  lower  between  the  mount  and 
plane.   As  a  first  approximation  we  will  neglect  the 
resistance  between  the  mount  and  plane  as  small  com- 
pared with  the  upper  recoil  resistance.  Let 

MR  =  mass  of  upper  recoiling  parts 

MC  =  mass  of  lower  recoiling  parts 

ra  =  mass  of  the  projectile 

vo  =  the  muzzle  velocity  of  the  projectile 

V  =  the  initial  velocity  of  the  recoiling  parts 

v  =  the  velocity  of  combined  recoil 

Then,  during  the  impulsive  action,  neglecting  the  mass 
of  the  projectile,  we  have, 

T 

for  the  projectile  /  Pdt  =  mvo     (1) 


T        T 

for  upper  recoiling  parts  /   Pdt  -  /  Rdt  =  MV  (2) 

o        o 

Where  F  is  the  vertical  reaction  between  the  upper 
and  lower  recoiling  parts.         T 

How  R  is  a  finite  force,  .*.  /  Rdt  -  0,  if  t  is 

o 

very  small.   Further  the  displacement  of  the  upper 
and  lower  recoiling  parts  inappreciable,  since 

T  T 

/   Vdt  =  0  and  /  Fdt  =  0  respectively 
o  o 

Hence,   nvo  =  MpV  with  no  appreciable  displace- 
ment of  either  the  tipper  or  lower  recoiling  parts  and 
no  moraetitura  imparted  to  the  lower  recoiling  parts. 
During  the  recoil,  after  the  impulsive  action,  we  have 


326 


T 

for  the  upper  recoiling  parts  /  Rdt=MR(V-v) 

o 

T 

for  the  lower  recoiling  parts  /  Rdt=Mcv 

o 

hence,  the  combined  velocity  of  the  system  when  the 
relative  recoil  between  the  upper  and  lower  recoiling 

parts  ceases,  is 

MRV 

v  =  T. : — 


If  the  mutual  recoil  reaction  R  between  upper  and 
lower  recoiling  parts  is  made  constant,  then 

v"  c  n   V 

R  =  Mc  -—    or  T  = — -    where  T  is  the 

time  of  the 
relative  recoil.   The  relative  displacement  Z  is, 

,V+v        v"       V 
?        99 

£t  &  6 

Substituting  for  T,  we  have 

McMR    V* 

Z  3  — — —    •     for  the  relative  displacement 
MR  +  Mc   2R 

The  relative  displacement  can  also  be  obtained 
from  a  consideration  of  the  energy  relations  in 
the  recoil.  We  have 

V          1        X  _* 

T)  =  —  **R(V  -v  )   for  the  upper  recoil- 
parts 

V  i     a 

-T~  T)     =  -  Mcv        for  the  lower  recoil— 


parts 


Subtracting: 
RZ  =  J  MR(Va-v2)-  f  Mcv* 


that  is  the  energy  of  recoil,  j  MRV  = 

is  dissipated  in  friction  and  throttling  (RZ)  and 


327 


remainder  is  the  kinetic  energy  of  the  combined 
masses.   Now  since,      M  v 
-      R 

=  MR+*c 
we  have  »       * 


i  MRMc   t 
•  «  MR+MC 

Therefore  as  before,  the  relative  displacement  becomes 


MR+MC   2R 


ELEMENTARY  RELATIONS.     During  the  travel  of  the 

projectile  in  the  bore  of  the 
gun,  neglecting  for  a  rough 
approximation  the  mass  of  the 
powder  gases,  a  mutual  reaction 

is  created  "by  the  powder  gases  between  the  gun  and 
projectile,  which  generates  equal  momentum  in  both 
projectile  and  gun  provided  no  extraneous  forces 
are  exerted  on  the  gun.  The  resistance  of  the  recoil 
brake  is  very  small  compared  with  the  powder  force, 
therefore  its  momentum  effect  is  negligible.  After 
the  projectile  leaves  the  bore,  further  expansion  of 
the  gases  take  place  and  the  reaction  due  to  the 
momentum  generated  in  these  gases  causes  an  addition* 
al  increment  in  momentum  of  the  gun.  This  additional 
momenta  is  commonly  known  as  the  after  effect  of  the 
powder  gases. 

Assuming  free  recoil  of  the  gun,  if 
m  =  mass  of  projectile 
M  =  mass  of  the  gun  or  recoiling  parts 
P  =  total  powder  reaction 
v  =  absolute  velocity  of  projectile 
V  =  absolute  velocity  of  gun  in  the  recoil 
u  =  relative  velocity  of  projectile  in  bore 
then  during  the  travel  up  the  bore  /  Pdt  -  mv  =  MV 
but  u  =  v  +  V  for  the  relative  velocity  of  the  pro- 


328 


jectilc,  hence  m(u-7)=MV  and  the  velocity  of  recoil 

becomes 

.  m   .  mv 

V  =  ( )u=  — 

m  +  M  N 

Since  m  is  snail  compared  with  M,  we  are  not  great- 
ly in  error  in  assuming  u  =  v  in  approximate  cal- 
culations. 

At  the  end  of  the  travel  of  the  projectile  up 
the  bore,  we  have  mv 

and  7( 

After  the  projectile  leaves  the  bore  if  P  =  the 
reaction  exerted  by  the  gases,  then 

**  /  it 

/   Pdt  =  M(7f  -  V0)  =  nv   where  v  =  the  mean 

tg velocity  of  the  gases 

"m"  after  expansion.   For  a  first  approximation 
v  will  "be  assumed  a  function  of  the  muzzle 
velocity  v0  and  we  will  place  BV  =cvQm 

Hence  MVf=(m+cin)vo.   For  computations  c  will 
be  taken  equal  to  2.3.   The  energy  of  free  recoil 
becomes 


hence 

T.  =  i  — 

M 

How  the  recoil  brake  exerts  a  resistance  R  through 
a  recoil  displacement  b,  "hence 

Rb=  -MV*  roughly, 


and 

R  =  

2M.b 

The  recoil  reaction  R  is  a  measure  of  the  stressed 
condition  of  the  carriage  and  very  often  for  a  given 
carriage  m,  u,  vo  and  b  may  one  or  all  be  changed. 
To  compare  the  recoil  reactions,  we  have  for  the 

sane  gun,  t  t 

Rt   (•t*cit)  v0i  b, 


,,_ 

and  for  R  =R  =R,  then  —  = —  where  c  = 

\    (-.*«,)•  v*, 

2.3  approx.,  and  for  bt  =  bf,  =  "b,  then 

Rt   (in^+cl^)3  v0± 

r~  a  ~, ., — : —    where  c  =  2.3  approx. 

R,    (»,+co>t)*  v«, 

These  equations  are  important  in  order  to  estimate 
with  a  given  change  in  the  ballistics  of  a  gun,  the 
necessary  change  in  either  the  recoil  or  recoil 
brake  reaction. 

The  energy  of  recoil  nay  "be  expressed  as 

m  +cu,  t     _.   a  . 

E  =  1  r  (m+cm)  v0  } 

M 

r  jM-f  mvo)   very  roughly 


m 

=  -  (muzzle  energy  of  the  projectile)  (approx.) 

M 

Therefore,  to  decrease  the  recoil  energy  M  should  "be 
made  as  large  as  possible.   Since  further 


The  recoil  reaction  varies  inversely  as  the  recoil- 
ing mass,  and  therefore  to  decrease  R,  M  s"hould  "be 
made  large. 

EFFECT  OP  POWDER  GASES     The  effect  of  the  pow- 
ON  THE  RECOIL.          der  gases  on  the  recoil  may 

be  considered  during  two 
periods:-  (1)  while  the  pro- 
jectile travels  up  the  bore, 

(2)  after  the  projectile  leaves  the  "bore  and  the  ex- 
pansion of  the  gases  ta"kes  place.   In  either  case  an 
approximate  assumption  is-  necessary  in  order  to 
represent  the  phenomena  with  sufficient  simplicity. 

During  the  travel  of  the  shot  up  t"he  bore  it  will 
be  assumed  that  the  gases  expand  in  parallel  lamina, 
and  the  motion  of  any  differential  lamina  to  be  a 
linear  function  of  the  distance.   from  the  "base  of 


330 


the  "bore  to  the  lamina  in  question,  that  is 

i  v  +  V 

v  =  c  s  +  c   where  c  =»  -  V  and  c  =  

u 

v  =  velocity  of  projectile 
V  =  velocity  of  recoil 
u  3  travel  of  projectile  up  the  "bore 
hence  with  free  recoil 

TO    I 

mv  +  2  -  v  =  MnV  during  the  travel  up  the  bore 
u 

but  „ 

»   i  m  ,u   i .    m(v-V) 

2-v  =-/  vds=  — 

u  u  0           2 

The  equation  of  momentum  of  the  system  during  the 
travel  up  the  bore  becomes,  therefore, 

-  (Y~V)  (m*0.5S)v 

•  v  +  m  — - —  =  MV    or    V  = 


MR+0.5m 

Further  since  the  relative  velocity  of  the  projectile 
is 

~  =  v  +  V   then,  l»+0.5D(?r-  V)  =(M+0.55)V 
at  at 

du 
therefore     (m+0.5m  )T~ 


and  for  the  displacement  of  recoil  in  terms  of  the 
relative  displacement  of  the  projectile, 

(m+0.5l) 


M+V+I 

If 

P  =   the    reaction  of   the   powder  gases   on   the 

"base   of   the   projectile 
Pfc3   the   reaction  of  the   powder  gases   on   the 

base   of   the  "bore  of   the  g"un 
then,    for   the  powder  gases,    we  have 

I  d(v-V)          5  d_v  _  ra  d_V  ( 

^>"    "   2  "     dt  Z  dt  ~  2  dt 

for  the  motion  of  the  recoiling  parts  in  free  recoil, 


331 


Pb  -  *R  ST  ia4**'  °ai^*a'1  ^" 
and   for   the  motion   of   the  projectile 


If  tbe  gun  moves  backwards  a  displacement  X,  while 
the  projectile  moves  forward  an  absolute  displacement 
x,  then 

X  *  /  Vdt,    x  =  /  vdt   (4) 
Prom  (2)  and  (3)  in  (1), 

dV  dv  .  i.  dv  _  £  dV 
MR  dt  "  dt  2  dt  ~  2  dt 
hence 

(Mp+0.51)—  =  (B+0.5  1)^      (5) 

Integrating,  we  have  as  before, 
(MR+0.5ii)V=(m+0.5i)v  (6) 

and. 

(MR+O.SijX'dn+O.SSJx  (7) 

For   the   relative  displacement 

u  =  /l      (V+v)dt      or  du  =    (v+V)dt 
o 


du     (NR+0.65)d-u 
V+v 


.      v        .        ....         r      / 
.    .    X  -  /        Vdt   =  /      ( 


_-)  du 
o  o       Mp+tn+n 

hence 

m+0 .  5n 

X  *  ..          as  was   obtained  by  direct   sub- 

Mo  +ra+m 

stitution   of  displacements. 

With   a   constant  powder   pressure  during    the   travel 
•up   the  bore,    the   time  of  travel  becomes, 

2u0        2u0(MR+0.5in)        (2WR+m)    UQ 

*   =  — —   =  i  =  -••^^-•^ 

v+7        (t 


Actually  since  the  powder  reaction  varies  during  the 

travel  up  the  bore, 

/U°  °*u  .          fU°  d" 
o  K        0 

Since  m  and  if  are  always  small  compared  with  Mp,  we 

have 

,uo  du 
t  =  /   7—       very  closely 

o 

The  relation  between  P^  and  .P  may  be  obtained  as 
follows: 

m  A ( v— V ) 

at    *  p 


1  .  dv 

«  '  T~     approximately 

2  dt 


hence 


or 


0.5— 


Since  however  the  linear  motion  of  the  powder  gases 
is  an  assTaraption',  we  "have  more  accurately, 

dv 
P-JJ  =  (ID  +  Bi)  -—       where  for  a  first  approximation 

B  =  0.5 
The  rngaa  powder  pressure  lies  "between  P^  and  P  hence 


Pffl  =  (1  +  B  -SL-)  P  where  for  a  first  approximation 
8"  =  0.3 

ELEMENTARY  ENERGY     The  Kinetic  energy  of  the  pow- 
RELATIONS.         der  gases  may  also  be  considered  a 
summation  of  the  elementary 
energies  of  the  differential 
lamina.   Assuming  the  gases  to 

move  up  the  bore  in  parallel  lamina,  with  the  velocity 
of  any  lamina  a  linear  function  of  the  end  velocities 
and  neglecting  the  velocity  of  the  gun  as  relatively 


333 


small  compared  with  that  of  the  projectile,  we  have, 
for  the  kinetic  energy  of  the  powder  gases, 

i  =  total  mass  of  powder 

gas 
u  =  travel  up  "bore  of 

projectile 


where 


•  i    s 
hut  v  =  - 


•0  yS.O 


v  =  velocity  of  any 

given  lamina 
s  =  distance  from  "base 

of  Tbore  to  lamina 


in  question 


1  /m  *  * 

V  »  (3  )V 

.  0+  :  ' 

The  Kinetic  energy  imparted  to  the  recoiling  parts 

IS  22 

1  (m+Q.Sm)  v 

ED=  —      —""-"•• "••• 

2  M 

"•.'•  qi"  ,•  '    ** 

tooien*Qxe  i>Ai  \£  trevij  ex  »ic 
Further  if, 

W  =  the  potential  energy  of  the  gases  at  any 

instant 
P^  =  the  total  reaction  exerted  on  the  treech 

of  the  gun 
P  =  the  total  reaction  exerted  on  the  base  of 

the  projectile 
X  =  the  displacement  of  the  gun  in  the  direction 

of  its  movement 
x  =  the  displacement  of  the  gun  in  the  direction 

of  its  movement 
Q  =  heat  lost  in  radiation 

J  =  the  mechanical  equivalent  of  heat 
then,  the  equation  of  energy  of  the  powder  gases  he- 
comes 

-  PbdX  -  Pdx  =  d(Bp+W)+  JdQ 

that  is  the  external  worfc  on  the  powder  gas  system 

goes  into  kinetic,  potential  or  configuration 

energy  and  lost  heat  energy.  The  above  equation  may 

"be  written  -dW  =  P^dX  +  Pdx  +  dEp  +  JdQ 


Further  since  PbdX  =  d(J  (m*°'5i)'  V>  ) 


Pdx  =  d(  mv*  ) 


We  have, 

-  „  .  4  [  i  (("*°-Sii>\  .*|>,']«  JdQ 

*     M  3 

The  work  done  on  the  system  may  "be  represented  by  an 
equivalent  force  Pm  acting  through  a  distance  cor- 
responding to  the  travel  of  the  projectile  up  the 
"bore,  then  -  dW  =  Pm  du  +  JdQ  and  since  du  =  dx,  very 
closely,  we  have  t 

r((n+O.Sm)        m  ,    dv 

Pm  =  t  -  *  m  +  r  1  v  — 

M  3     du 

Thus  the  equivalent  mass  of  the  system  gun,  projectile 
and  powder  gases,  referred  to  the  displacement  up  the 
bore  is  given  "by  the  expansion, 

(w+0.5i)        i 

M«  =  -  +  m  +  - 

R          3 

RECOIL  AND  BALLISTIC     The  recoil  reaction,  say,  when 
MEASUREMENTS.         the  gun  is  mounted  on  a  ballistic 

pendulum  and  the  reaction  of  Vhe 
projectile  when  fired  into  a 
ballistic  pendulum,  differ  by 
fhe  reaction  caused  by  the  ex- 

pansion and  consequent  acceleration  of  the  powder. 
Obviously  the  snaller  the  charge  the  wore  closely 
would  the  swings  of  these  pendulums  "be  alike. 

BALLISTIC  PENDULUM  -  QUM  HOUHT8D  OB  PEHDULUM. 

(a)     When  the  powder  charge  is  very 
small,  we  have  an  equal  impulsive 
action  on  the  projectile  and  gun. 
If 

d  =  the  distance  from  the  axis  of  rotation  to 


335 


the  center  line  of  the  bore. 

M  »  the  mass  of  the  pendulum  and  gun  combined, 
k  =  radius  of  gyration  about  the  axis  of  sus- 
pension. 

9  -  angle  turned  by  the  pendulum 
h  »  distance  from  the  center  of  gravity  to  the 

axis  of  suspension. 

Then  in  consequence  of  the  mutual  impulse  during  t~he 
fire,  mv.d  =  Jfk*w  and  the  initial  angular  velocity 

is.  therefore. 

mv.d 
w  =  — — —     (rad/sec) 

Hk»  _, 

d  e     «h 

The  subsequent  motion  is  given  by,  — —  =  -  *j-  sin  8 

Integrating, 

.de.i        2gh 

W  =^COS  e  +  c 

de 

when  6   =  0,    cos  9=1   and— — =   w 

dt 

therefore 

t'       2gh 

c  =  "    -v~ 

and 

,d8    »        2gh  2 

(-rH      =  — ~  (cos   6-1)    +   w 

U  U  1C 


Q  w 

At   the  maximum   swing    (—  —  •)    *  0,    and   6  =   9Q,   hence 

Q  t 

, 

-  cos  e) 


0 


This  is  immediately  evident  from  the  equation 

of  energy,  since    ,,  ,• 

Hk  w 

=   Mgh(l  -  cos   e      ) 

2  o 

e0 

The   cliord   of   an   arc   radius    "c"    is    1   =    2c   sin-— 

2 

e 

Further  since,  1  -  cos  8Q  =  2  sin -— "- 
So    mv.d 


336 


M  It   „  9o 

hence  v   =  --  2  sin  — 

n  d  2 

M  k   1 

=  —  —  —  v^gli 
mac 

whic"h   means   the   velocity  of   the  projectile   approximate- 
ly.       The   radius   of  gyration   may  readily  "be  obtained 
experimentally  by  noting   the   time   of   swing. 

(b  )  When   the   powder   charge   is   com- 

para"ble  with  the  weight   of   the  pro- 
jectile,   we  have   to  consider   the 

additional   momentum  generated  by 

the    powder  gases. 

Assuming  the   center   of  gravity   of   the  powder  mass   to 
have   a  mean  -velocity  equal   to   one-half   the   velocity 
of   the  projectile,    we  have 

(1)  during   the   travel    up   the  bore, 

• 
(m+—  •)  v        as   the   momentum 

m 

generated  in  the  gun. 

(2)  after  the  projectile  leaves  the  lore 
we  have  an  additional  impulse  p  due 
to  the  expansion  of  the  gases. 

Hence  the  equation  for  the  motion  of  the  "ballistic 

pendulum  becomes,          -          2 

d[(m-»-  —  )v+p]  =  Mk  w 

2/pT     eo   1  IvTO 

but  w  *  —  •—  sin  —  »  --  *  — 
k       2    c  i( 

1  ,     Mkl 
hence  (•+  —  )v+p»  —  •  /gh 

2  cd 

If  now  we  repeat  the  experiment  with  the  powder  gases 
as  done  in  the  experiments  on  the  Ballistic  Pendulum 
"by  Dr.  Hutton,  we  "have 


7V°  *  p  * 

where   obviously     V     is  greater   than  v, 


337 


S         Mk(l-I0) 

Subtracting,  we  have  mv+-~-(v-v  )= *•  /gh 

2     °     cd 

i          M  (l-l«)fc 
or  v-  --   (v_-v)=  —• 


2m          in    cd 

To  account  for  the  powder  gases  experimentally,  Dr. 
Button  proposed  measuring  with  and  without  the  pro- 
jectile as  follows: 

Mkl 

rov+p1    =  — T  /gh  with   the   projectile 

cd 


hence      (i-i)1c 


o 

—  -  - 
m    cd 


The  previous  expression  indicates  this  expression  in 
error  "by  the  amount   _ 


which  for  small  charges  is  relatively  small  tut  for 
large  charges  may  be  appreciable  and  therefore  can- 
not be  neglected.  As  an  approximation,  however,  in 
ordinary  tests,  the  method  of  Dr.  Button  is  suf- 
ficiently accurate,  for  the  measurement  of  the 
velocity  of  the  projectile. 

BALLISTIC  PEMDOLUM  -  IMPDLSE  OP  PROJECTILE 

The  "ballistic  pendulum  serves  as  a  valuable 
mechanical  means  of  measuring  the  velocity  of  the 
projectile  though  this  method  has  been  discarded  in 
modern  practice.   The  dynamics  involved  is  worthy 
however  of  consideration  in  the  general  recoil  pro- 
"blem. 

The  time  of  penetration  is  sufficiently  s~hort 
for  no  appreciable  movement  of  the  pendulum. 

Let  d  =  the  perpendicular  distance  from  the  axis 
to  the  line  of  penetration  of  the  pro— 


338 


jectile. 
J  »  the  distance  from  the  axis  to  the  position  of 

the  projectile  when  the  penetration  ceases. 
B  »  the  angle  between  "d"  and  "J" 

Then,  the  impulsive  moment  of  the  projectile  Mp  equals 
the  change  in  its  angular  momentum,  hence 
Mp  *  mv.d  -  mJ*w  and  the  corresponding  reaction  on  the 
•pendulum  "becomes  M_  =  Mkaw.  Therefore  mvd  =  (m"k*+-mJa  )w 
or  mvJ  cos  B  =  (Mk*+mJ*)w.  The  initial  energy  of  the 
system  consisting  of  the  pendulum  and  projectile  is, 
therefore 

« 


w 

and  the  worlc  done  by  the  weights  in  the  movement  to 
the  maximu  swing,  "becomes,  Mgh(l-cos  6)+mgJ[cos  B- 
cos(9  -  B)]  hence,  from  the  principle  of  energy,  we 
have, 

j(Mk«+mJ«)w«=Mgh(l-cos  e)+mgj[cos  B-cos(6  -B)] 
If  B  *  0,  the  equations  reduce  to  mvJ  =(Mk*+mJ*)w 

(Mk«+mJ*)w*=2(Mgh-mgJ)(l-cos  8Q) 
Combining  these  equations  and  noting  that 

eo 

1  -  cos  9=2  sin*  ——  ,  we  have,  for  the  initial 
2 

velocity  of  impact  for  the  projectile, 
2 


+  mJ»)(Mh  +  mJ)g  ]  sin  - 
•J  2 

GENERAL  THEORY  OF    In  the  preceeding  paragraphs  the 
RECOIL.          theory  of  recoil  was  greatly 

simplified  by  assuming  the  powder 
period  to  "be  of  such  short  duration 
as  to  be  in  the  nature  of  an  im- 

pulsive action,  and  therefore  the  momentum  of  recoil 
being  generated  practically  instantaneously.   In  tha 
theory  of  impulsive  forces,  we  may  neglect  finite  forces 
such  as  the  resistance  to  recoil  since  the  time 
of  action  is  negligible.   Further  the  displacement 
in  an  impulsive  action  is  entirely  negligible.  This 
method  gives  fairly  accurate  results  for  long  recoil 


339 


but  when  fbe  recoil  is  shortened  the  results  "by  this 
method  of  computation  are  only  very  approximate. 

Fortunately  due  to  considerable  progress  made 
in  interior  "ballistics  of  late,  the  powder  reaction 
can  be  determined  as  a  function  of  time  and  displace- 
ment up  the  bore.   It,  therefore,  "becomes  a  finite 
force  and  the  recoil  problem  during  the  powder  period 
can  be  treated  with  a  considerable  degree  of  accuracy. 
Let  Pjj  =  the  total  powder  reaction  on  the  breech 
in  Ibs.   Its  line  of  action  is 
necessarily  along  t"he  axis  of  the  bore. 
B  =  the  total  braking  due  to  the  hydraulic 

and  recuperator  pulls. 
R  -  the  total  friction,  (guide  and  packing 

frictions)  in  Ibs. 
K  -  the  total  resistance  to  recoil. 
Hr=  the  mass  of  the  recoiling  parts 
¥r=  the  weight  of  the  recoiling  mass  in  Ibs. 
X  =  the  displacement  of  the  recoiling  mass 
from  battery  in  the  direction  of  the 
glides. 

0  =  the  angle  of  elevation  of  the  g"un 
a  -  the  angle  of  the  guides  constraining  the 
recoiling  mass  with  respect  to  the 
horizontal . 

From  the  theory  of  energy,  we  have  the  fundamental 
principle: 

The    work   done    on    the    system    consisting   of    the 
recoiling    part.g    "hy    t.hft    pnariar    gagfts    must    ptQiial     tha 
work    dons    on    fha     system   T">y    t^ift    t. otal     ra gi  gt. anr*.?*.    tin 
recni  1      for     t.hfi     *»nf.i  rr»     rftnni  1         an  nf./°     t."ha     enftrrfy     r>f     t.Vis 
^ysteyn     ar     t,Tia    bstfinning     and     °nd     rvf     rf.r'.DJ]     i  g     7.»rr>- 

Froro  this  theorem  we  may  prove  that  with  a  re- 
sistance to  recoil  action  throughout  the  powder 
period,  the  energy  which  the  powder  imparts  to  the 
recoiling  mass  whan  free  is  always  greater  than  the 
energy  which  must  be  developed  by  the  brafce  in  the 
recoil.   The  greater  the  resistance  to  recoil  during 
the  powder  period  the  greater  this  deviation. 


340 


In  the  following  proof  the  time  effect  of  the 
powder  gases  during  free  and  constrained  recoil  is 
assumed  the  sane  or,  in  other  words,  the  powder 
reaction  is  regarded  the  same  for  any  given  time 
whether  the  recoiling  mass  is  contrained  or  free. 
Theoretically  of  course  due  to  the  slightly 
different  motions  in  the  two  cases,  the  notion  of  the 
powder  gases  themselves  will  be  slightly  different 
and  therefore  a  slightly  different  reaction  on  the 
breech  clock  in  the  two  cases.   Since,  however,  the 
difference  in  motions  is  so  small  and  the  powder  re- 
action so  great,  we  may  entirely  neglect  this  fact 
and  assume  the  powder  force  to  be  entirely  a  function 
of  time  and  quite  independent  of  the  slightly  different 
motion  in  constrained  and  free  recoil. 

Supposing  the  gun  to  recoil  along  the  axis  of  the 
bore  as  is  usually  the  case,  the  total  resistance  to 
recoil  evidently  may  be  expressed  as:  K  =  B+R-Wr  sin  0. 

Therefore,  the  equation  of  motion  for  the  re- 
coiling mass  for  constrained  recoil,  becomes, 

dV  dvf 

Pv  -  K  =  ID  -T—   and  for  free  recoil,  we  have  Ph=fflr  - 

dt 

Integrating  for  any  given  time,  evidently,  V  <  V^ 

The  work  done  by  the  powder  for  contrained  recoil  is 
therefore  less  than  with  free  recoil,  since 

*i  t 

Pt,V  dt   <  /  l  Fb  Vf  dt   where  tt  =  the  total  time 

00  of  the  powder 

period.   Kow  the  work  done  by  the  brake  must  equal  the 
work  done  by  the  powder  gases  in  constrained  recoil, 

hence, 

b         t 

/   Kdx  =  /   PV  dt 


b        t, 

/   Kdx  /   PbVf  dt 


341 

t  b 

but  /  *  PbVfdt  =  j  rarVf   therefore  /   Kdx  <  j  mr  V*{ 
o  o 

that  is,  the  braking  energy  or  rather  the  work  done  by 
the  resistance  to  recoil  provided  the  braking  is  effect- 
ive during  the  powder  period,  is  always  less  than  the 
free  energy  of  recoil.  When,  however,  no  braking 
resistance  acts  during  the  powder  period,  the  work 
done  by  the  resistance  to  recoil  or  braking  energy 
must  equal  the  free  energy  of  recoil.   Therefore,  for 
a  given  length  of  recoil,  the  recoil  reaction  is  re- 
duced by  maintaining  a  resistance  during  the  powder 
period  in  a  twofold  way: 

(1)  due  to  the  fact  that  gun  recoils 
over  a  greater  distance,  (i.  e.  the 
displacement  during  the  retardation 

and  in  addition,  the  displacement  during 
the  powder  period), 

(2)  due  to  the  fact  that  the  braking 
energy  is  always  less  than  the  free 
energy  of  recoil. 

In  the  design  of  a  recoil  system  it  is  there- 
fora,  highly  desirable  to  maintain  a  large  resistance 
to  recoil  during  the  powder  period  and  thus  effective- 
ly to  reduce  the  required  braking  and  the  consequent 
stresses  set  up  in  the  carriage,  as  well  as  to  give 
better  stability  to  mobile  mounts. 

GENERAL  EQUATIONS     (1)     When  the  direction  of 
OF  RECOIL.  recoil  is  not  along  the  axis  of 

the  bore.     Consider  the  re- 
coiling paris  to  be  constrained 
along  guides  or  an  inclined 

plane  making  an  angle  "a"  with  the  horizontal,  and  the 
axis  of  the  bore  to  make  an  angle  0  with  the  horizontal. 

Neglecting  the  reaction  of  the  projectile  normal 
to  the  bore,  as  small  compared  with  the  other  reactions, 
we  have  for  the  equation  of  motion  for  the  recoiling 
mass . 


342 


Pv,  cos  (0+a)  -  B  -  R  -  Wpsin  a  *  m,  -2.JL   (D 

dt« 
hence 

Pt,  cos  (  0+a)-B-R-Wrsin  a)  dt  »  mp  dv 
and 

/Pb  cos  (0+a)dt  -  /(B+R+Wrsin  a)dt  =  mrv 
but  the  powder  force  is  measured  by  the  rate  of  change 
of  momentum  imparted  to  the  recoiling  mass  when  free, 
that  is        dVf 

P"  '  'r  IT 
hence  Pb  cos  (0+a)dt  »  mr  cos  (0+a)d  Vf 

Substituting  in  the  above  equation,  we  have 
mrVf  cos  (0+a)  =  /(B+R+Wr  sin  a)dt  =  mrV     (2) 
When  the  resistance  to  recoil  is  constant, 
K  =  B  +  R  +  lfr  sin  a  =  a  constant,  and  we  have 

Vf  cos  (0  +  a)  t  =  V  (3) 

Integrating  again,  we  have, 

Kt* 

/Vf  cos  (0  +  a)  dt  -  — —  =  X 

2mr 

which  gives  the  displacement  from  battery  of  the 
recoil  during  the  powder  period,  but 
/Vf  cos  (0  +  a)  dt  =  E  cos  (0  +  a)  which  is  the 
component  displacement  for  free  recoil  in  the 
direction  of  recoil. 

The  constrained  recoil  at  the  end  of  the  powder 

period,  becomea  ? 

KT 


X  =  E'  =  E  cos  (0+a)  -  T  —   (4) 


2mr 

and  the  corresponding  velocity  at  the  end  of  the 
powder  period,  becomes, 

vr  *  Vf  max.c°s^+a>  ~—  <5 

where  T  is  the  time  of  the  powder  period. 

Proa  the  energy  equation  in  the  motion  from  the 
end  of  the  powder  period  to  the  end  of  recoil,  we 


343 

have  ~  mpvp  =  K(b-xt)   hence 


j  mr[Vfcos(0+a)  -—  ]   =  Kb-K[E  cos(0+a)-  j^-]   6) 
Expanding  and  simplifying,  we  have 


K[b-E  cos('0+a)+VfT  cos(0+a)=  jmrVf  cos*(£j+a)] 

hence 

t   ,,a     x  ,  .,   . 

-mrVf  cos  (0+a) 

K  =  b-(E-VfT)cos(0+a)  (7) 

or  in  terms  of  .the  component  reactions, 

1      2'       2* 

-,mrVf  cos  (0+a) 

B+R+W.  sin  a  =  -  (?') 

b-(E-VfT)cos(«f+a) 

where     WVQ+  4700  ^ 

Vf  =  -   from  the  principle  of  linear 
wr        momentum. 

E  =  total  free  movement  of  gun  during  powdei 
period. 

T  =  total  time  of  powder  period, 

To  deduce  E  and  T  we  proceed  as  follows:  (See  Chapter 
II)    Calculate 


rf  Z1  Pm   ti    fa   27  ^N*  -.  i 

b  3  uo[(  TS  r~  ~  1}  i  /(1  "  T^  r~)  ~  1  ] 

16  Pe  16  Pe 

where  p^  =  max.  powder  pressure  X  area  of  bore 


and  also, 


~  b      »  Pbm  : 


then  compute  — 

wv0+  4700  w  2 

Vf  =  -  -  -  j  V= 

L             *a                J  ... 


344 


where  w  -  weight  of  projectile 

w  »  weight  of  charge 

wr  =  weight  of  recoiling  parts 

VQ  =  nuzzle  velocity 

The  time  of  the  travel  of  the  projectile  up  the 
"bore  and  the  time  during  the  expansion  of  the  powder 
gases  are  respectively: 

b  .        2u   u  2(Vfl-Vfo)   wr 

*o  -  ;  <*'3  log  -  *  -  *  8)   tlo  .  —  ^—  _ 

3   uo 
*  -  --  approx. 

vo 
Therefore  the  powder  period,  "becomes  T  =  to+tlo 

The  free  recoil  displacement  during  the  travel  up 
the  bore,  and  during  the  expansion  of  the  gases  are 
respectively: 

u0(*+0.5w) 


Therefore,  the  total  free  movement  of  gun  during  the 
powder  period,  becomes,   E  =  X  fo  + 


MOTE:    In  the  above  and  further  formulae  the  units 
employed  are  : 

displacement  in  feet 
velocity  in  feet  per  second 
force  in  pounds 
mass  in  pound  units 

With  a  void  in  the  recoil  cylinder  during  part 
of  the  powder  period,  equation  (7)  becomes  slightly 
modified. 

Let  S  =  length  of  void  in  recoil  cylinder 

tg  =  time  of  free  recoil  to  end  of  void 
Neglecting, 

R+Wr  sin  a  as  small  compared  with  B,  ws 
find  K  =0,  until  distance  S  is  reached  in  the  recoil. 


345 


Therefore  we  "have 


=  E  cos  (0  +  a)  - 


K(T-ts) 
2*. 


cos(0+d)  - 


K(T-ts) 


(8) 
(9) 


where  T  -  time  of  total  powder  period.    Substituting 
(8)  and  (9)  in  the  energy  equation, 

-,  mrvr  =  K(b-xt) 
and  simplifying,  we  have 

1        2f      2' 

-  mrVfCOS  (0+a) 
K  =  b-CE-Vf(T-ts)]cos(0+d) 
To  evaluate  ts,  t"he  time  of  recoil  with  void,  we  have 


t.  =  -(2.3  log  ^-  +~  +  2) 
a         D    D 


. 


where 


(w+-£-)cos(0+a) 


-  D±  /l-  ~) 
16  =  P 


(11) 


Chater  II. 


(12)   S  "being 

the  length 
of  void. 


See 


III 


vo  =  muzzle  velocity  in  feet 

uo  =  total  displacement  up  "bore  in  feet 

pm  =  max.  powder  pressure,  Ibs.  per  sq.  in. 


64'4  u 


(15)  mean  powder 
pressure, 
Ibs.per  sq. 
in. 


346 


Ab  »  area  of  bore  of  gun. 

If,  ho*ever,  the  length  of  void  corresponds  to 
•  displacement  greater  than  the  recoil  displacement 
for  the  projectile  to  travel  up  the  bore  of  tbe  gun, 
we  have, 

b         2u0   a0 

t.  -  -  (2.3  log  —  *  —  *  2)  +  t 

a          oo 


or  approx. 


>    (16) 


3  uo   Ai 

5  T.  *  *• 


where  tt  is  obtained  froa  the  solution  of  the 
cubic  equation, 


C-|j  (  -ji  -     -  )  +  Vfo  }  cos  (0  *  •)  -  x;  -  0   (17) 


n  here 


rfo 


:  V, 


wv0  +  4700 


2(Vf,-Vfo) 


32.2 


and  X^ 


ao  cos  (0  +  a) 


(18) 


27   a    u 
also  P0b  *  4"  *   (b+u)»   d-12  PB  V         <19> 

Powder  reaction 
on  breech  when  shot  leaves  muzzle. 


COI8TBAIS1D  VILOOItT  Of  BBOQILi 

(1)     During  powder  pressure  period. 
Knowing  R  from  the  previous  formulae,  tbe  con- 
strained velocity  of  recoil  nay  be  computed  from  the 


347 


free  velocity  curve  as  follows: 

From  equation  (3)  we  have,  V  *  V*  cos  (0+a)  -  — 

«r 

and  the  corresponding  displacement 

*  Kt* 

X  »  /   ?f  cos  (D  +  a)  dt  -  ~- 

o  4mr 


Kt* 
»  X*  ees  (0  +  a)  

X  QH 

smr 

Thus  we  see  the  free  velocity  curve  of  recoil  both 
against  time  and  displacement  of  free  recoil  is  re- 
quired in  order  to  compute  the  constrained  velocity 
curve. 

The  free  velocity  curve  during  the  powder  period 
is  divided  into  two  periods,  (1°)  the  velocity  of  free 
recoil  while  the  shot  travels  up  the  bore,  and  (2°) 
the  velocity  of  free  recoil  during  the  expansion  of  the 
powder  gases  after  the  shot  has  left  the  muzzle. 

Lednc's  formula  gives  us  a  means  of  computing  (!•) 
while  Vallier'a  hypothesis  serves  for  the  computation 
of  (2«). 

From  Lednc's  formula,  we  have,  during  (1°)  of  the 
powder  period, 

v  -  r^ —  (20) 

b  +  u 

"b          2u   u 
t  -  5  (2.3  log  -5-  +-B-  +  2)      (21) 

where 

a  *  travel  up  the  bore  in  feet 
QO  a  travel  up  the  bore  to  muzzle 

v  *  corresponding  velocity  of  projectile  in  the 

bore  of  the  gun  (feet  per  sec) 
vo  *  muzzle  velocity  of  projectile 

t  *  corresponding  time  of  the  travel  in  seconds. 

.  t  27  Pm         /I 27  Pm.a ~  . 

*-..t(rB--»±/a-Ie-)  -13 


348 


pm  »  max.  powder  reaction  on  base  of  projectile 

B 

wv0 

pe  3  -•••••  *  mean  reaction  on  base  of  pro- 
0   jectile  during  travel  up  bore. 

a  3  (b^o)  12 

Farther  from  elementary  dynamics,  (see  Chapter  II) 

(w+f)v 

Vf  •  — (22) 

wr 

(   *\ 
2 

X    a   •   i      .» 

r 
or  approx.    ^ 

2 
X,  -  (23) 


where  w  >  weight  of  projectile  in  Ibs. 

v  *  weight  of  powder  charge  in  Ibs. 
wr  >  weight  of  recoiling  mass  in  Ibs. 

The  procedure  therefore,  to  compute  the  free 
Telocity  carve  against  time  and  displacement  during 
period  (1°)  is  as  follows 

(a)  Compute  b  and  from  it  a, 

(b)  For  various  displacement  up 
the  bore:  compute  v  and  t. 
(Equation  20  and  21). 

(c)  Then  from  equations  (22) 
and  (23),  compute  V  and  X  . 

Arrange  the  data  in  a  table  with  corresponding  values 
of  V  ,  X  and  t. 


349 


Prom  these  values  the  constrained  velocity  carve  during 
(1°)  nay  be  computed  from  equations  (3)  and  (4).  Front 
Vallier's  hypothesis,  we  have,  during  (2°)  of  the  pow- 
der period,  for  the  total  pressure  on  the  breech 
Ft,  -  Pob  -  C(t  -t0)   (Valuers'  hypothesis) 


where 


C 


t  -  t0       tt  -  t0     2(Vf,  -Vfo)mp 
hence 


Now,  from  elementary  dynamics,  the  change  of  momentum 
along  the  axis  of  the  bore,  becomes, 

t 
/   Pfc  dt  =  mr(Vf  -  Vfo)  (25) 

*o 

Substituting  (24)  in  (25)  and  integrating,  we  have 


obo         4mr(Vf.-Vfo)  r° 

fying,   we 

have,  for  the  free  velocity  of  recoil, 


0 
Vf  -  Vfo  »  -   (t-t0)(l  - 


The  corresponding  displacement  of  free  recoil,  along 
the  axis  of  the  bore, 

t 
Xf  *  Xfo  +  /  Vf  dt  (27) 

*o 

w 
where       w  +  - 

Xf0  »  -  u0       uo  =  total  travel  up  the 
r  bore  in  feet. 

if  if.*/*  v{0dt^A,-t0><.t  -  4..P(vt..Tfo)  A*-* 


350 


Simplifying,  the  displacement  of  free  recoil  for  tine 

t,  becomes, 


The  following  initial  values  and  constants  are  to  be 
substituted  in  equations  (27)  and  (28). 


fo  "  ~ —       *fo 

wv^  +  4700  w 


t0  -   (2.3  log     *    4  2) 


3  uo 

»  -  —  approximately. 
3  v 


27  B    .      ,    27 


Pm  »  aax.  powder  reaction  on  base  of  projectile. 

a 

wvo 
^«  *  ft  7  '  J  "    *  mean  reaction  on  base  of  projectile 

daring  travel  op  bore. 

27  t   a 
P0b  "  7-  b   .   .^'  1.13  Pffl  «  reaction  on  breech  of 

gun  when  the  shot 
leaves  the  muzzle. 
The  procedure,  therefore,  to  compute  the  free 


351 


velocity  curve  against  time  and  displacement  daring 
period  (2°)  is  as  follows: 

(a)  Compute  Pm,  Pe,  and  then  b 
and  a  as  before. 

(b)  Compute  Pob,  to, (Vfi-Vfo  and 

Xfo. 

(c)  Then  from  arbitrary  time  intervals 
between  tt  =  T  and  to  compute  from 
equations  (26)  and  (28)  Vf  and  Xf 

Arrange  the  data  in  a  continued  table  as  in  (1°)  with 
corresponding  values  of  Vf,  Xf  and  t. 

Prom  these  values  the  constrained  velocity  curve 
during  (2°)  may  be  computed  from  equations  (3)  and 
(4). 

MAXIMPM  V1LOOITY  OF  COJ8TBAIIHD  BIOOILt 

The  condition  of  maximum  velocity  of  constrained 
recoil  is  when  the  powder  reaction  exactly  balances  the 
resistance  to  recoil,  since  before  this  condition  the 
recoiling  mass  is  accelerated  and  immediately  after  it 
is  retarded. 

Hence  Pb  cos  (0  +  a)  -  K  =  0 

t  -  t. 


2.r(V  -Vto)     « 

Hence  the  time  at  the  maximum  velocity  of  constrained 
recoil,  is  obtained  from  either  of  the  following 
equations:- 


t~t 


Solving  for  t,  we  have 


(30) 


352 

2m(Vf i-Vfo)[Pob  cos(0+a)  -  K] 

or  t. *  Pobt0  (30') 

Pob  cos  A  *  a 

Substituting  tm  in  (26)  and  (28),  we  have, 

Vfm  -  vfo  +  -«7  ^m-to)  t1  ~  4mr(Vft-vfo) 
and 


(32) 

where  Vfm  and  Xfm  are  the  free  velocity  and  displacement 
corresponding  to  the  maximum  constrained  velocity 'of 
recoil. 


BEOAPITDLATIQH  Of  FORMULAE  FOB  PRIBCIPLE  PEBIQDS  DURIKQ 
PQITDER  PRESanHE  P1BTQD. 

In  the  constrained  velocity  curve  daring  the 
powder  period,  we  have  the  following  important 
points: 

(a)  Velocity  and  displacement 
of  the  recoil  when  the  shot 
leaves  the  muzzle. 

(b)  Maximum  velocity  and  its 
corresponding  displacement  of 
recoil  and  time. 

(c)  Velocity  time  and  corresponding 
displacement  at  end  of  the  powder 
period. 

Given  data: 

»r  =  wt.  of  recoiling  parts. 

VQ  =  nuzzle  velocity. 

w  =  weight  of  projectile. 

w  *  weight  of  powder  charge. 

u  *  total  travel  of  shot  up  bore. 


353 


Pm  =  max.  powder  reaction  on  "base  of  shot. 

P^  =  max.  intensity  of  ponder  pressure  assumed 

X  area  of  bore, 
b  *  length  of  recoil. 

INTERIOR  BALLISTIC  OOMSTAHT8  BgflPIBED  FOB  VELOCITY  CUBVB; 

Pe  *  mean  average  powder  reaction  on  base  of  shot  * 

2 

!I°- 

2gv0 
B  =  twice  abeissa  of  max.  pressure, 


27  Pm   ,,    /7   27 

•re?--"  i^-u 


*  max.  velocity  of  free  recoil 


a  velocity  of  free  recoil  -  shot  leaves  muzzle 
w  +  0.5  w 


pob  *  total  pressure  on  breech  when  shot  leaves 
muzzle. 

* 


4     (B+u0)» 
tQ  *  time  of  recoil  while  shot  travels  to  muzzle 

B  ,        2u   v        3  uo 
=  -  (2.3  i0g  --  +  -  +  2)  -  -  ~  approx. 
a  «  vo 

tt  *  time  during  the  expansion  of  gases  after 
shot  leaves  muzzle. 

m  a(vf.-vfo)  j^ 

pob      < 


354 


t  *  T  »  time  for  total  powder  period 

""""'*.  *  »,„  "  "  c" 

Xfo  »  free  movement  of  gun  while  shot  travels  to 
muzzle 

u0(w+0.5  I)   w+0.5  w 
»         -   =  — — —  u0   approx. 

\f io  =  free  movement  of  gun  during  expansion  of 
powder  gases. 


Total  free  movement  of  gun  during  powder  period 

B  •  *fo  *  xf'o 

K  »  resistance  to  recoil:  t  *  angle  of  elevation: 
a  *  angle  of  plane  of  guides  with  horizontal, 


»   «•     *  f*  \ 
-  «rVfi  cos   (0+a) 

b-(E-Vf ,T)cos(0+a) 


VKLOCITY  AMP  PT 8PL AGKMR MT8   AT   PERIODS 


(a).(b)  and  fe). 

At  Period  (a): 

V0  and  X0  »  the  constrained  velocity  and  dis- 
placement in  recoil  for  period  (a) 
when  shot  leaves  the  muzzle. 

Kt0 
V0  »  Vfo  cos  (0+a)  

Kt« 
X0  »  Xfo  cos  (0+a)  -  ^— 


355 


At  Period  (b): 


tm  »  time  at  max.  velocity  of  constrained  re- 
coil. 

K(T-t0) 


Pobcos(0+a) 

m  and  Xfm  =  velocity  and  displacement  of  free 
recoil  at  the  instant  of  maximum 
velocity  of  constrained  recoil. 

pob  ^ob^m"*©) 


xfm  -  xfn  *  tVfo  *  « — (tm~to)  ~  » — TTT- 

10   2mr         6mr(Vf 
V-  and  X_  =  maximum  constrained  velocity  and  cor- 

m       in       ••• *i^ ^ ^ 

responding  displacement  of  recoil. 
Ktm 

x^       \         "I 

Vm  =  Vfm  cos  (0  +  a)  -  — 


xm  =  xfm  cos 

At  Period  (c): 

V  =  Vr  =  constrained  velocity  of  recoil  at  end 
of  powder  period. 

Xt  -  EP  a  corresponding  displacement  of  constrained 
recoil  at  end  of  powder  period. 

Kt 

V  »  V_  =  Vf i  cos  (0+a) 

mr 

Kt2t 

X,  «  EP  =  Xfi  cos  (0+a)  » 

oin_ 

f 


356 


UNITS  TO  BE  EMPLOYED  IN  THE  ABOVE  AND  FURTHER  FORMULAE; 

BRITISH  8Y8TIM  MITRIO  SYSTEM  METRIC  SYSTEM 
QBAYITATIONAL  9R AV IT  AT  I  01 AL  GRAVITATIONAL 
UNITS.  OMITS.  UNITS. 


Displacement   in  feet  -  ft.    in  meters*n   in  centimeters' 

cm 

Telocity        in  feet  per      in  meters      in  centimeters 
see. -ft/sec.     per  sec.>      per  see  * 
m/se  c .         em/se  c . 


Force 


pounds  -  Ibs  .   Kilograms  =    Kilograms 
kg.  kg. 


Pressure 
Intensity 


lb  s  .  sq.  in, 


Kg. per  sq. 
cm. 


Kg.  per  c  q. 
en. 


Pressure 
Area 


. inches        Sq.cn. 


Sq.  em. 


Mass 


Lbs/g   (£-52.2)   Kgs/g 


Kgs/g 
£  =  981 


Ti»e 


Seconds  =Seo.    Seeonds=8ec*  Seconds  -  Sec. 


OOJ8TRAH1D  VKLOCITY  COBVK: 

(2)     During  Retardation  Period  of  Recoil. 

After  the  ponder  period  the  recoiling  mass  is 
brought  to  rest  by  the  resistance  to  recoil.  The 
recoiling  mass  then  reaches  the  extreme  out  of  battery 
position. 

At  the  beginning  of  the  2°  period  of  recoil,  the 
recoiling  mass  has  an  initial  velocity  Vt  *  Vri  and 
an  initial  displacement  from  battery  Xt  *  Er. 


357 


A  V 
Prom  the  equation  of  motion,  we  have  K  »  -  m_V  — 

dX 

Integrating,  between  the  limits  X,  to  any  given 
displacement  X,  and  between  corresponding  velocity 
V,  to  Vg  we  have 

X  Vx 

/    K  dX  -  -  mr  /    V  dV  (33) 

X  V. 


Hence, 

retardation  period  of   the   recoil.      Hence 


K(X-X  )»  -    which  is  the  equation  of 
2 

energy  during  the 


2K(X-X. ) 


(34) 


rar 

A  simpler  and  more  direct  form  for  computing  the 
constrained  velocity  during  the  2*  period  of  recoil  is 
as  follows: 

We  have,  as  before  K  dx  *  -  mr  V  d  V 

Integrating  between  the  limits  X  and  b  in  the 
displacement  and  Vx  and  o  in  the  velocity,  we  have 

m.Vx 
K(b-X)  -  -J-^-  (35) 


/2K 
—  (bH 


Hence  Vx  -  /  —  (b-X)  (36) 

ror 

showing  that  the  velocity  during  the  retardation 
period  is  a  parabolic  function  of  the  displacement. 

It  is  to  be  especially  noted  that  a  characteristic 

of  a  constant  resistance  to  recoil  is  a  parabolic 

function  of  velocity  against  displacement. 


GENERAL  EQUATIONS  OP  RECOIL     In  the  previous  formulae 
CONTINUED.-  VARIABLE  RESIST-  the  resistance  to  recoil 
ANCE  TO  RECOIL.  was  assumed  constant 

throughout  the  recoil. 

It  is  however  often  de- 
sirable for  stability  to  decrease  the  resistance  to  re- 


358 


coil  in  tbe  out  of  battery  position  and  thus  partially 
compensate  for  tbe  decreased  stability  due  to  the 
moment  effect  caused  by  tbe  overhang  of  the  recoiling 
mass  in  the  out  of  battery  position. 

With  a  variable  resistance  to  recoil  it  is  customary 
to  maintain  a  constant  resistance  during  tbe  powder 
period  and  thence  decrease  tbe  resistance  proportional 
to  tbe  displacement  to  the  out  of  battery  position, 
with  a  given  arbitrary  slope  "m".  See  Chapter  III. 

Let  KQ  *  the  constant  resistance  during  the  pow- 
der period. 

Vt  and  Vr  *  tbe  velocity  of  constrained  recoil 

at  the  end  of  the  powder  period, 
b  =  total  length  of  recoil. 

Then  tbe  equation  of  tbe  resistance  to  recoil 
against  displacement  of  recoil  becomes, 
K0  »  constant,        from  0  to  Xx  or  Br 

(37) 
K  *  Ko  —  m(X-X  )      from  X^  to 

Further,  g  j 

Vt  »  Vp  »  Vf •  cos(0+a) 2.  (38) 

mr 

K0T« 

X  *  Er  »  B  cos  (0+a)  -  (39) 

o  rn  — 

Now  fron  tbe  equation  of  motion  of  the  recoiling  parts 
during  the  retardation,  we  have,  K  dX  »  -  mr  V  dy 

Integrating  between  limits,  X,  and  b:  and  V,  and  0, 


K  dX 


d  V 


Hence  ^  m  y* 

[I0  -  m(X-Xt)J  dX  >  -j-i 

Integrating,  we  have  for  the  energy  equation, 


359 


K0(b-Xt) 


»(b-Xt)   mrVi 


(40) 


Substituting    (38)   and   (39)   in   (40)    and   neglecting   the 

terra          j/*T4 
m    ^o1- 

in  the  expansion  as  small,  we  have 


ra  "oj 
2   4 


f»rVf. 
<  - 

2 


cos*(0+a)+  ^  [-b- 


] 
> 


,  T« 
b-B  cos(0+a)+  Vf  cos(0+a)  T  -  -  —  [b-E  cos(0+a)] 

o  ffl— 

Thus  from  the  ballistic  constants  E  and  T,  together 
with  the  length  of  recoil,  maximum  free  velocity  of 
recoil  and  any  given  arbitrary  slope  "»",  the  re- 
sistance to  recoil  maintained  constant  during  the  pow- 
der period  may  be  computed. 

Substituting  KQ  in  place  of  R  in  the  proceeding 
formulae  during  the  powder  pressure  period  enables  as 
to  compute  the  retarded  velocity  curve  daring  the 
powder  pressure  period. 

During  the  retardation  or  second  period  of  recoil 
we  have,  K  dx  »  -  mrV  dV 

Integrating,  from  the  displacement  x  to  the  end 
of  recoil,  we  have 
b          v 
/  Kdx  *  mr  /  J   Y  dV 


therefore  /  (Ko  ~  «(X-Xt)]dX  » 

x 

Hence 


mPV* 


K(b-X)  - 


•X 


mX,X 


and  simplifying,   we  bave 


360 


"rVl 

[K0  -  -   (b+X^X,  ' 
2 


Hence  j = 

:(K  -  -  tb+X  -  2X)](b-X) 

O 


(42) 


where  as  before, 

m  »   the    arbitrary  slope  of  resistance  to  recoil. 

o    m* 

X  a   E  cos   ()0  +    a)  -  iv- — 
*mr 

GENERAL  EQUATIONS  OP     When  the  direction  of  recoil 
RECOIL  -  Cont.        is  along  the  axis  of  the  bore, 

(a)  Constant  resistance  to 
recoil  throughout  recoil, 

let  K  »  B  +  R  -  Wr  sin  0  =  total  resistance  to  recoil 
B  *  total  braking  R  =  total  friction 

E  3  displacement  in  free  recoil  during  powder 

period. 

T  =  corresponding  time  for  free  recoil, 
then  for  the  motion  of  the  recoiling  parts, 

dV  T  pb      KT 

Pb  -    '  mrdT  (  ~f  dt  *  m7  3l  Vr 

but  as  before  /  ~  dt  =  Vf  =  max.  free  velocity  of 

recoil,  hence 

KT* 
and  the  corresponding  displacement,  X  *Er  »  E  —  

0IH«B 

After  the  powder  period,  from  the  equation  of 
energy, 

—iH-V..  »  K(b  —  X.) 


KT 


*  / ..  Ri.a  ..  /.  Rlv 

-  m_(Vf )      '   K(b  -  E   *  - — ) 

m  P  c  ffl  j. 


361 


t     Tf« 


*"r*f 
and  simplifying,  we  have  K  »  (43) 

This  equation  obviously  is  a  special  case  of  equation 
(7)  since  when  (0  +  a)  *  0,  cos(0  +  a)  =  1  and  a  =  -  0, 

(b)     Variable  resistance  to  recoil. 

The  resistance  to  recoil  as  before  is  assumed 
constant  during  the  powder  pressure  period  and  thence 
to  decrease  uniformly  consistent  v/ith  stability,  that 
is  with  a  stability  slope  as  given  in  Chapter  III  on 
stability. 

At  the  end  of  the  powder  period,  we  have  for  the 
constrained  velocity  of  recoil  and  corresponding  dis- 
placement, 


At  the  end  of  recoil,  the  resistance  to  recoil  be- 
comes KQ  -  mfb-Sr)  where  m  *  the  stability  slope 
(See  Chapter  III). 

The  mean  resistance  from  the  end  of  the  powder 
period  to  the  end  of  recoil,  becomes, 
2K-m(b-Er) 

-  —  =  K0  -  ?  <b-*r> 

2 

and  from  the  equation  of  energy  of  the  recoiling 
mass,  we  have 

K00>-Br)  -  2  (b-Ep)*  -  7  mr  V*      (46) 


Substituting  the  values  of  Er  and  Vr  from  (44)  in 

m 

2 


(46)  and  neglecting  the  term   _*_» 

m  K  T 


we  have, 

mV  *  m(b-B)a 


2  -r 


362 


This  equation  obviously  is  a  special  case 
of  equation  (41)  since  when  (£J  +  d)  »  0,  cos(j0+a)  *  1 
and  d  »  -  0.  .*.. 

(c)     Dynamic  equation  of  recoil 
during  powder  period. 

Since  during  the  ponder  pressure  period,  the  re- 
sistance to  recoil  is  assumed  constant  even  with  variable 
recoil,  we  have,  therefore,  the  same  dynamic  equation 
with  either  variable  or  constant  resistance  to  recoil 
during  the  powder  period. 

Dividing  the  powder  period  into  two  intervals  tQ 
and  tx  -  to  while  the  shot  travels  up  the  bore  and 
during  the  expansion  of  the  powder  gases  after  the  shot 
has  left  the  bore,  respectively,  we  have 

(1)     During  the  travel  up  the  "bore, 


/  ^ 

(48) 


Kt 

u  -  —  (49) 


8         2u   u 
and  t  -  -(2.3  log  —  +  -  +  2)  (50) 

a        B    B 

Thus  V,  X  and  t  are  functions  of  the  parameter  u.   The 
ballistic  constants  a  and  B  have  been  determined 
previously  in  this  Chapter  as  well  as  in  Chapter  III 
in  "Interior  Ballistics". 


363 


When  the  shot  reaches  the  muzzle, 


/     \ 

("*") 

2 


Kt 


(B  +  u0)  mp 


(61) 


w 


(52) 


o    o 
and  t  -  -  (2.3  log  —  +  —  +  2  ) 


3  uo 
*  —  —  approx 

^     o 


(53) 


(2)     during  the  expansion  of  the  powder 
gases,  we  have 


au 


uo   where  u  =  the  total 

travel  up 

the  "bore,  the  dynamic  equation  of  recoil  during  this 
period  becomes, 


ir  a         dV 
-  K  »  rar  - 

dt 


(54) 


2 


where  Pob   »          B2   --  -  -      1>12   pm   (See  Chapter   III) 

4  \B    +    u  -  / 


Integrating,    we   find 


(55) 


l     "O- 


364 


Hence  V  -  V.+  -  -  -  [1  --  -  ]  _  _  (t-t0)(56) 

2(tt-t0) 

The  corresponding  displacement  is  obtained  by  integrat- 
ing equation  (55) 


2        6(t-tQ) 


V  d(t-t) 


+  mr  V0  (t-t0)  +  Const.  (57) 

Now  mr  /  V  d(t-t0)  =  m(X  -XQ).   Hence  where  t  =  tQ, 
X  »  X0  and  const.  *  0.  Simplifying  (57)  we  obtain 

for  the  recoil  displacement  during  the  second  period 
of  the  powder  period, 


X  =  X0  +  V0(t-t0) 


2m 


(58) 

To  obtain  the  maximum  restrained  recoil  velocity  and 
corresponding  displacement,  we  must  equal  the  total 
powder  reaction  to  the  total  resistance  to  recoil, 
that  is  Pb  -  K  »  0 

Pobd  --  )  -  K  »  0   where  PO^  »  the  pressure  on 
\~t-o  tnc  Creech  when 

the  shot  leaves 
the  muzzle. 

t  =  total  time  to  maximum  restrained  recoil 
velocity,  hence  solving  for  tm,  we  have 

• 
ta  »  tt  -  —  —  (tt  -  to  ).   Substituting  in  equation 

ob  (56)  and  (58)  we  have 

Ppb^nrV  .    tm"to   .    K 


2(tt-tQ)    m 


365 


(t  -t  )*  P°b(tm'to)  M-  *•"*» 
*  3 


2^        *<tt-t0) 

(60) 

At  the  end  of  the  powder  period, 
t  »  tt  =  T  and  X  »  Er  and  V  =  \  and  substituting 

again  in  eq.(5c)  and  (58),  we  have 

(t     -tn    )        K 

Vr    -   V0    *  Pob   -i-  --  ~   (tt   -t0    )  (61) 

2mw  fflr 


>* 


(62) 

(d)     Dynamic  equation  of  recoil 
during  the  retardation  or  the 
pure  recoil  period. 

(1)  constant  resistance  to  recoil: 

Since  the  total  resistance  to  recoil  is  constant, 
the  velocity  must  be  a  parabolic  function  of  the 
displacement  of  the  recoil, 

Prom  the  principle  of  energy,  we  have, 

M  V*  /~~~t    \~ 

K(b-X)  »  — —   hence  V  »  /2  

2  mr 

(2)  Variable  resistance  to  recoil 
The  resistance  to  recoil  out  of  battery,  becomes, 


k  *  K  -  m  (b-E  +  - — )   where  K  = 
2mr 

•  "~*~ 

The  average  resistance  to  recoil  in  the  displacement 
b  -  X,  becomes 

k  *  \  (b-X) 
From  the  energy  equation,  we  have, 


366 


k(b-X)+  I  (b-X)'  -  j-j»r  V* 

I 
2 


/Kb-XHTc*  •  (b-X)] 


v 


COMPONENT  REACTIONS  OF       Let  K  *  total  resistance 
THE  RESISTANCE  TO  RECOIL.  to  recoil. (Ibs. or 

Kg) 
B  =  total  braking.  (Ibs. 

or  Kg) 
R  »  total  friction  to 

recoil. (Ibs.  or  Kg) 

Ph  =  reaction  of  hydraulic  brake. (Ibs. or  Kg) 
Pv  *  reaction  of  recuperator. (Ibs. or  Kg) 
px  *  hydraulic  brake  pressure. (Ibs/sq. in)  or  (Kg/I2 
A  3  effective  area  of  hydraulic  brake  piston,  (sq.in. 

or  m  ) 

py  »  recuperator  pressure.  (Ibs/sq.in)  or  (Kg/m  ) 
Ay  *  effective  area  recuperator  piston. (sq.in  or  m  ) 
Vo  "  initial  volume  of  recuperator. (cu. ft.  or  m  ) 
X  *  recoil  displacement. (ft.  or  m) 
Sf  *  final  spring  reaction.  (Ibs)  or  (Kgs) 
So  =  initial  spring  reaction. (Ibs)  or  (Kgs) 

The  total  resistance  to  recoil  then  becomes 
along  the  bore  along  special  guides 

K  *  B   +  R  -  Wr   »in  0  K»B+R+Wr  sin  6 

where  0  »  angle  of  elevation,  9  *  angle  of  guides. 

Now  in  systems  where  the  hydraulic  brake  is  independent 

of  the  recuperator  system,  B  »  Pn  +  Py 

In  systens  where  the  brake  and  recuperator  are 
connected  B  »  Pj, 

For  independent  systems 


PVJ(T T~\~]    ^or  pneumatic  recuperators 

"" 


367 


Sf-so 
v  »  S0  +  ( )  x  for  metallic  recuperators 

o 


and  Pyi»  1.3  Wr(sin  0,  +  u  cos  0m)  approx. 


hence 


KAS 
-7-    where  c  =  -7- 

wx  "x 


V     k      * 
vo   *    cv 


V  —A* 

ro 


)   *  ~r~    for  pneumatic  re- 

"v 

cuperators. 


sf-so       cv* 
B  =  S0  +  ( )  x  +  — 7—  f or  metallic  re- 

x  cuperators. 

For  systems  where  the  hydraulic  brake  and  recuperator 
are  directly  connected, 

,   KA' 

P-PV  =  — —  where  c  =  —— 

*       Mf*  M* 

wx  "x 


pviA  *  Pyj  *  1.3  Kr(sin  0  +  u  cos  0)  approx. 


therefore 
P  = 


•  » 
o  v 


pA 


'  i  * 
c  AT 


since 


and  c  A  »  C   hence  B 


cv 


~ 


which  is  an  equation  of  exactly  the  sane  form  as  for  a 
system  where  the  recuperator  is  independent  of  the 
hydraulic  brake. 


368 


The  general  equation  for  tbe  resistance  to  recoil 
"becomes, 

(a)     when  the  recoil  is  along  axis  of  tbe 
bore: 

a 

cv 
*  ~  —  +  R  -  Wr  sin  t,  for  pneumatic 

V 

recuperators. 


K  -  S0  +  (  —  -  -  )x  +  -yr  +  R  -  Wr  sin  0,  for  metallic 


recuperators, 
(b)     when  tbe  recoil  is  along  special 
guides: 

V          * 

O  CV 

K  =  pvi(VTT4 — )     *  ~T~  *  fi  *  wr  sin  e»    for  pneumatic 

V  f^  AX  W  y 

recuperators. 

SfSo 


+  -7-  +  B  +  Wr  sin  6,  for  metallic 


recuperators. 

a 
CV 

K  »  -7-  +  R  +  ffp  sin  9,   for  gravity  mounts. 


GENERAL  EQUATIONS  OP     The  function  of  the  recuperator 
COUNTER  RECOIL.      is  to  return  tbe  recoiling  mass 

into  "battery.   The  stability  of 
a  mount  in  counter  recoil  is 
greatest  at  the  beginning  of 

counter  recoil  and  least  at  the  end  of  counter  recoil 
or  when  the  gun  enters  tbe  battery  position.  To 
prevent  shock  and  unstableness  as  the  gun  arrives  in 
battery  it  is  necessary  to  introduce  some  form  of 
counter  recoil  buffer  towards  tbe  end  of  counter  re- 
coil.  Very  often  a  buffer  resistance  of  varying 
amount  is  introduced  throughout  the  counter  recoil. 
In  addition  we  always  have  the  resistance  of  tbe 
guides. 

Without  a  recuperator  tbe  recoiling  mass  must  be 


369 


returned  to  battery  by  the  gravity  component  due  to 
the  inclination  of  the  guides  with  the  horizontal. 
If  this  inclination  is  small,  the  gravity  component 
does  not  greatly  exceed  the  friction  and  thence  a  very 
elementary  buffer  may  be  used,  the  return  velocity 
being  always  small. 

Let  KV  3  total  unbalanced  force  in  counter  recoil. 


Fy  =  recuperator  reaction. 

=  variable  orifice  for  counter  recoil  buffer. 


By  =  counter  recoil  buffer  resistance. 


Ay  =  effective  area  of  recuperator  piston 
py  =  pressure  intensity  in  the  recuperator 
cylinder. 

pa  =  pressure  intensity  in  the  air  reservoir. 
R  =  total  friction  of  counter  recoil. 

During  the  accelerating  period  of  counter  recoil, 

we  have 

dv 

Kw  =  HD  v  —    and  during  the  retardation 
dx 

dv 

Kv  »  -  mR  v  —  - 

dx 

During  the  acceleration  Kv  is  necessarily  always 
smaller  than  the  total  resistance  to  recoil,  "hence 
during  the  acceleration  counter  recoil  stability  is 
of  no  consequence.   During  the  retardation,  if 

d1  =  the  distance  from  front  hinge  or  wheel  contact 
with  ground  in  a  field  mount,  to  the  line  of 
action  of  the  total  resistance  to  recoil. 

L  =  horizontal  distance  between  front  and  rear 

supports  of  mount. 
Ls  =  horizontal  distance  from  rear  support  to 

center  of  gravity  of  total  system  with 

recoil  parts  in  battery. 

b  =  total  length  of  recoil. 


370 


*3  =  weight  of  total  mount. 

Then,  for  a  gun  recoiling  along  the  axis  of  the  bore 
during  the  retardation,  Kvd'  ^  ltg(L-Ls)  +Hr(b-X)cos  0 

and  the  minimum  stability  occurs  when  the  gun  enters 
"battery,  that  is  Kyd '  ^  WS(L-LS).  The  stability  slope 
of  counter  recoil,  becomes      ^  cos  ^ 

m1  *  — — — — 
d1 

To  consider  the  components  of  the  total  resistance  to 
counter  recoil,  we  have  three  classifications: 

(1)  recuperator  systems  independent  of 
the  hydraulic  bralce  and  with  no  throttling 
between  the  air  and  recuperator  cylinders. 

(2)  recuperator  systems  independent  of 
the  hydraulic  brake,  with  throttling 
between  the  air  and  recuperator  cylinders. 

(3)  recuperator  cylinders  connected 
directly  with  the  brake  cylinder.   In  all 
systems  an  independent  buffer  may  be  in- 
troduced in  either  the  recuperator  or 
brake  cylinder  front  end.   In  certain 
types  the  buffer  acts  as  a  plunger  brake 
within  the  piston  rod  of  the  recoil 
brake. 

Then, 

(1)  for  recuperators  independent  of  the  "bralce 
cylinder  and  with  no  throttling  between  ths 
air  and  recuperator  cylinders, 

Kv  =  Fv  -  B'X  -  Wr  sin  0  -  R       (1) 
when 


0 


Vo  =  initial  volume 


1.3  Wr'(sin  0+0.3  co«  0)  approx. 

(2)  for  recuperators  independent  of  the 
bralce  cylinder,  with  throttling  between 
the  air  and  recuperator  cylinders, 


371 


pvAy  -BjJ  -  Wr  sin  0  -  R 


where          «  2 
c  v 

pv  "  pa     2 (WQ  =  constant  orifice  usually) 
wo 

V^ 


Fyj.  =  1.3  Wr(sin  0  +  u  cos  0) 

hence 

•  2 
c  v        i 

KV  =(Pa 2~'Av  ~  Bx  "  wr  sin  &  ~ 


and  since   »  2 


"o 


wo 

then  the  equation  reduces  to  same  form  as  (1),  that 
is  Ky  =  Fv  -(Bi  +  BJ  )  -  Wr  sin  0  -  R, 

(3)  for  recuperators  directly  connected  with 
the  recoil  bralce  cylinder, 
Ky  *  pyA  -  B£  -  Wr  sin  0  -  R 

where          »  » 
c  v 

P\r  =  Pa    W2         ("x  =  variable  orifice 

by  buffer  rod  on  a 
floating  piston  in 
recuperator  or  air 
cylinder.) 


rt«  ^V0-A(b-x)  J    -  Fv 
F  .  =  1.3  Wr(sin  0  *  u  cos  0)  hence, 

Ktt  =  F  -  (B*  +  B")  -Wr  sin  0  -  R, 
v    v 

"  2 
«     C  V 

where  Bx  =  ~ —  Av 

A        ,,7          V 

"x 

which  is  again  an  equation  of  same  form  as  (1). 


372 


The  general  equation  of  counter  recoil,    therefore, 
becomes 

*V  ~(&x  +  Bj)  -  Wr  sin  £J  -  fl  »  mg  v  -— 


where 

9      t 

a 

i 

L/n  Y  * 

|       y 

Bx 

"  c     ~ 
»x 

2gc"w»* 

o" 

DAjv* 

» 

*        V 

Bx 

»  *      "  * 

1  c       .« 

2gC         w  „ 

«•» 

CALCULATION  OP  RECOIL     It  is  often  convenient  to 
CURVES.  calculate  the  retarded  velocity 

curve  against  displacement, 
especially  when  the  resistance 
to  recoil  is  not  made  constant. 

In  all  cases  we  have  seen  the  resistanc-e  to  recoil  is 
in  general  a  function  of  both  the  displacement  and 
velocity  of  recoil,  that  is  the  recuperator  component 
of  the  recoil  resistance  is  a  function  of  the  displace- 
ment, whereas  the  bralce  component  is  a  function  of  the 
velocity  and  the  variation  of  the  throttling  orifice. 
Hence  K  =  f(xtv)  and  the  dynamic  equation  of  recoil 

18  dV 

Pb  cos(0  +  0)  -  K  »  WR  — -  or  when  the  recoil 

translates  in  the 

direction  of  the  axis  of  the  bore, 

K      dV 

To  measure  Pj,  we  may  consider  the  momentum  im- 
parted by  the  powder  gases  in  free  recoil,  then 


Pb   "   «R   aT 

or  J     Pfcdt  »   BR^f"^ft)     Therefore,    for 

li  the   same   interval 

of    time    (t-tt)   we  have 


373 


»R(Vf-Vfi)  cos  (9*0)  -  K  (t-tt)  «  «R(V-Vt) 

be  nee 

V  »  Vt+(Vf-Vft)cos(9+0) U-tt)  or  when  the  re- 

"R        coil  translates 
in  the  direction  of  the  axis  of  the  bore, 

R  * 

V  =  V  +(Vf-Vf.)  (»-t.)   Further  since  X  =  X  +/  Vdt, 

Bo  • 

t 

we  have 

t 
X  =  Xt*Vt(t-tt)+  /  Vfdt  cos  (9+0)-Vft(t-tt)cos(9+^)  - 


K  (t.t  )*     now  J  Vfdt  =  Xf-Xft   hence 
2«R  4t 


X  =  Xt  +  [t-Vf 

t 
(t-t  )   or  when  the  recoil  translates  in  the 

direction  of  the  axis  of  the  "bore, 

x  -  x^O^-v^Mt-^MXf-x^)-  Jj-  (tf-ta)* 

Therefore  the  velocity  and  displacement,  for  any  given 
interval  (tt-ts) 

(a)     along  guides  not  parallel  to 

the  "bore: 

vt»vt+(vft-vfl)cos(e+0)  -  —  (tt-tt) 

•R 


xt=xt+(?t-vftcos(e+0)](tt-ttMxft- 

(t,-tt)« 

(b)     alon^  guides  parallel  to  the 
axis  of  the  "bore: 


374 


After  the  powder  period  these  formulas  reduce  to 

w  -".-'t) 

•R 


2nR 

and  obviously  are  independent  of  the  direction  of  the 

guides  with  respect  to  the  axis  of  the  tore. 

Kt  +K2 
The  value  of  K  =  —  -  ,  which  may  he  closely 

approximated  by  a 

repetition  of  the  substitution  in  these  equations, 
since  from  the  first  substitution  we  closely  ap- 
proximate V2  and  thereby  can  determine  Ka=f(XaV  ) 
for  the  second  substitution. 


CALCULATION  0?  ACCELERATION,  TIME  AND  DISPLACEMENT 
PROM  A  GIVER  VELOCITY  CURVE: 


Recoil  and  counter  recoil 

velocity  curves  are  usually  drawn  experimentally  as 
functions  of  the  displacement  though  they  may  be 
drawn  as  well  as  functions  of  the  time.   The  customary 
method  of  obtaining  a  velocity  curve,  is  to  set  a 
tuning  fork  vibrating  and  allow  the  vertical  oscillations 
to  form  a  sinuous  curve  along  a  narrow  soot  covered 
strip  recoiling  with  the  gun.   Then  if  f  *  the  fre- 
quency of  oscillations  of  the  fork,  we  have  for  the 
time  of  one  oscillation,  T  =  -^-   If  n  =  the  number  of 

oscillations  for  an 
interval  Ax,  the  velocity  becomes, 

v  3  —  ,  where  At  =  nT    if  x  is  measured  in  inches, 
At 

'  '  13  ZT  (n/sec) 


375 


To  obtain  the  time  as  a  function  of  the  displace- 
ment, since  vdt=dx 

t  =  /  -  dx 

o  v 

1   *  1 

and  if  x  is  measured  in  inches,  t  =  —  /  -  dx 

v  o  v 

Hence  the  area  under  the  reciprocal  of  the  velocity 
curve  against  displacement  is  the  time  of  recoil. 

We  may  then  draw  the  velocity  curve  as  a  direct 
function  of  the  time  of  recoil. 

When  the  recoil  velocity  is  measured  as  a  function 
of  the  time,  the  acceleration  is 

dv 

•7-  =  the  slope  of  the  velocity  curve 

at 

When  the  recoil  velocity  is  measured  as  a  function 
of  the  displacement,  the  acceleration  is, 

dv 

v  ~  =  the  velocity  *  the  slope  of  the  velocity 

curve . 

=  t~he  sub-normal  of  the  velocity  curve. 
If  dx  is  measured  in  inches,  the  acceleration  is 

12  v  —  (ft/sec*) 

dx 

From  the  relations,  v=f(x)  and  t=/  -  dx  =/  — - — -  dx 

v       f(x) 

we  see  that  the  velocity  curve  may  "be  readily  expressed 
either  as  a  function  of  the  displacement  or  as  a  function 
of  the  time  or  both. 

CHARACTERISTICS  OF  RECOIL     From  Proof  Firing  T«sts, 

CURVES.  recoil  curves  are  obtained 

for  both  recoil  and  counter 
recoil.  From  these  curves, 
it  is  possible  to  determine 

the  variation  of  the  reactions  throughout  recoil  or 

counter  recoil. 


378 


In  the  analysis  of  curves  during  the  powder 
period,  since  the  mutual  relation  connecting  the 
variation  of  powder  force  and  the  retarded  recoil  is  the 
common  time,  it  is  necessary  to  express  the  forces, 
velocities  and  displacements  as  functions  of  the  time. 

In  the  analysis  of  curves  during  t~he  retarded 
recoil  and  counter  recoil  it  is  possible  to  express 
the  forces  and  velocities  as  direct  functions  of  the 
displacements  which  considerably  simplifies  the  work. 

(1)     Powder  Pressure  Period:  Recoil  along 
axis  of  "bore.  The  equation  of  recoil  is 

dV 

Pjj  -  K  =  Dp  T—   where  K  =  B+R-Wrsin  0 

With  a  given  velocity  curve,  the  velocity  and  displace- 
•ent  should  be  tabulated  as  a  function  of  the  time; 
then  for  any  interval  (ta~tt)  "e  have 

(vft-vfi>-  <Vvt)-L(tt-tt)  =  o 


=  o 


If  K  is  assumed  constant  or  found  to  be  constant  by 
brake  measurements  or  if  it  is  determinate  as  a 
function  of  the  displacement,  we  nay  evaluate  Vf   the 
free  velocity  of  recoil.   More  often  however,  the  free 
velocity  and  displacement  curves  can  be  evaluated  as  a 
function  of  the  time,  and  knowing  the  retarded  velocity 
and  displacement  curve  as  a  function  of  the  time  we 
may  calculate  the  resistance  to  recoil  from  the  above 
expressions.   Then        u  _y 

pb  =  "R      / 
la  -  *t 

and  ,4?  dV       dV 

pb-»R  jf '-      *here  "R  d"t  s  "R  v  al  '  "R0 

dV 
It  is  to  be  noted  that  Pv  and  -  Dp  r—  are  the  external 

d  t        — — ^^^— 


377 


recoil  forces  during  the  powder  period.   Further  P^  acts 
along  the  axis  of  the  bore  and  -  nR  £*-  acts  through 

the  center  of  gravity  of  the  recoiling  parts  parallel 
to  the  axis  of  the  bore  or  guides.   If  e  =  the  distance 

from  the  center  of  gravity  of  the  recoiling  parts  to 
the  axis  of  the  "bore,  we  have  for  the  external  reactions 
on  the  mount  a  couple  Pue  and  a  force  parallel  to 

dV 

the  axis  of  the  "bore,  Pv  -  mo  — —  =  K.  The  balancing 

dt 

forces  are  the  weights  and  reactions  of  the  supports. 
For  stability  the  moment  of  the  weights  about  the 
rear  support  must  exceed  the  moment  of  Pve  and  K 
about  the  rear  support. 

(2)  Retardation  Period:  Recoil  along 
axis  of  bore.  During  this  period,  we 
have  simply 

dV 


dt 


applied  through  the 
center  of  gravity  of  the 

recoiling  parts, 
«R  V  —  =  -  K  parallel  to  the  axis 

of  the  bore, 

which  together  with  the  weights  and  balancing  support 
reactions  are  the  external  forces  on  the  mount. 

It  is  to  "be  further  noted  that  since 


X  *  Ph  *  Pv  *  R  ~  wr  sin  ^  we  nave 

velocity  curve, 
dV 
Ph  »  -  oR  V Pv  -  R  +  Wr  sin  0 

f   dx 

V     k 

where  Pv  *  ?vi  I — )    f°r  pneumatic  recuperators 

V  -  A 
vo   *x 


R  =  0.25  lfr  cos  0  +  R_  approximately  where  Rp  =  estimated 

packing  friction. 


378 


(3)  Counter  recoil:    C'Recoil   along   axis 

of  bore. 

During  the  accelerating  period  of  counter  recoil,  the 
inertia  resistance  is  directed  towards  the  breech  the 

same  as  in  recoil.   Here 

dv 

Kv  =  mp  v  —  —    to  t"he  rear 
dx 

and  during  the  retardation  the  inertia  resistance  is 
directed  forward  and  "here, 

*v  -  *  »R  V  £ 

which  together  with  the  weight  of  the  system  and 
balancing  supporting  reactions  are  the  external 
forces  on  the  mount. 

Since  further,  during  the  retardation, 

dv          i 
-  mR  v  —  =  Fv  -  Bx  -  Wr  sin  0  -  R  we  have 


sin 


and  v 

o     » 
Fv  *  Fvf  [ ]    fof  pneumatic  recuperators 

Vft-A(b-x) 


and 

R  3  0.15  Wr  cos  0  +  Rp  approximately  where  Rp  = 

estimated  packing  friction. 

Since  critical  counter  recoil  stability  is  at 
horizontal  elevation,  C'recoil  curves  are  usually  ob- 
tained at  "horizontal  elevation.   Then, 

i  dv 

Bx»?v-R+»pV—   for  the  buffer  force  where 

^x   the  overturning  force  is 
dv 

•D  v  -—  along  the  axis  of  the  bore  forward, 
dx 

RECOIL  BBAKII0  WITH  A  CONSTANT  ORIFICE: 

As  a  first  approximation  we  will  assume  tne  re- 
cuperator reaction  not  to  vary  greatly  in  the  recoil. 


379 


Then  K  =   A  *    Bv     where   A  =   Pv   +  R  -  Wr  sin  t 

B  =  the  hydraulic  "brake 
throttling  constant. 

(1)     During  the  powder  period,  we 
have 

Pu  -  (A+BV*)»mR  — 
dt 

V  +V  •  . 


mR 
Expanding,  we  have 


which  is  a  quadratic  equation  of  the  form 
aV*  *  bV  +  c  =  0 

2  Z 

and 


-  b  ±  A)  -4ac 
V,  •     —      where  a  .  ^(t^J 

V  B 
b  »  1  *  -i-  (t,-tt) 

2  nn 


c  = 


,-t     -t- 

4«R  "R 


If   the    intervals   are   talcen  very  small,    then 


A  +BV* 


(t,-tt)*(vfl-vft) 


Then  solving  for  V2  we  may  repeat  with  the  expression 
V  +V 


V,  =  V±  --  -  (t.-tt)*(Vf,-7ft) 


380 


for  a  closer  approximation. 

The  displacement  is  obtained  from  the  expression, 

V  +V 


(2)     During  the  retardation,  we  have 
V  —  »  -  A  -  BV*  hence  dx 


•R  v  —  a  '  A+BV« 

•R    A*Bv 

Integrating,  we  have  X  -X  *—  loge 

28    ' 


In  particular  if  Xt  *  Eg  the  constrained  recoil  dis- 
placeaent  at  the  end  of  the  powder  period  and  vi  =  Vp 

tbe  constrained  recoil  velocity  at  the  end  of  the 
powder  period,  then,  for  any  displacement  x  and  recoil 
velocity  V,  we  have, 
»R  , 

x  "  ER  "  7*  loge 

or  with  common  logarithm, 


X  -  ED  «  1.15  —  log  - 
B      A+BV» 

when  V  =  0  the  length  of  recoil  "becomes, 

b  -  BR  +  1.15  —  log  (1+  -  Vj) 

B         A 

As  •  first  approximation,  we  may  take  Eg  *  E  the 
displacement  in  free  recoil  during  tbe  powder  period 
and  VR  *  Vf  the  maximum  velocity  of  recoil*  then 

h  -  B  *  1.15  -|  log  (1*  j- 


CO 

CHAPTER        VII 

CLASSIFICATION  AND  CHARACTERISTICS  OP  RECOIL  AHD  RE- 
CUPERATOR SYSTEMS. 

Recuperator  systems  nay  be  divided  into: 

(1)  Hydro  pneumatic  recuperator  systems 

(2)  Pneumatic  recuperator  systems 

(3)  Spring  return  recuperator  systems. 

(1)     With  hydro  pneumatic  systems,  we  have  two 

fundamental  arrangements:- 

(a)  The  hydraulic  brake  separate 
from  the  hydro  pneumatic  re- 
cuperator. This  requires  two  or 
more  rods,  a  brake  rod  and  a  re- 
cuperator rod.   Further  we  have 
in  general  two  or  more  cylinders, 
a  brake  cylinder,  a  recuperator 
cylinder  which  may  have  passage  way 
or  connection  with  an  air  cylinder. 
The  recuperator  and  part  of  the 
air  cylinder  is  filled  with  oil. 
The  oil  nay  be  in  direct  contact 
with  the  air  in  the  air  cylinder  as 
in  the  Schneider  and  Vickers 
material  or  it  may  be  separated 
from  the  air  by  means  of  a  float- 
ing piston  in  the  cylinder. 

(b)  The  hydraulic  brake  cylinder 
connecting  directly  with  the  recuper- 
ator cylinder.  The  oil  must  be 
throttled  between  the  recoil  and  re- 
cuperator cylinder,  and  thus  the  oil 
at  lower  pressure  reacts  usually  oa 

381 


382 


a  floating  piston  separating  the 
oil  and  air  in  the  recuperator 
cylinder.   To  augment  the  initial 
volume  for  the  air  in  the  re- 
cuperator an  additional  air 
cylinder  connecting  with  the  re- 
cuperator may  be  introduced.  Thus 
with  this  arrangement  we  have  from 
two  to  three  cylinders. 

(2)  With  pneumatic  recoil  systems,  we  have  usually, 

(a)  One  or  more  pneumatic  cylinders, 

according  to  a  satisfactory  layout. 
The  piston  compresses  the  air  directly,  no  oil 
or  other  liquid  being  used  for  transmitting  the  pres- 
sure. 

(3)  With  a  spring  return  system,  we  may  have 
various  arrangements: 

(a)  One  or  more  spring  cylinders 
separate  from  the  recoil  brake 
cylinder. 

(b)  With  small  guns,  the  spring  con- 
centric and  around  the  recoil 
brake  cylinder. 

The  potential  energy  or  the  energy  of  compression 
of  the  recuperator  during  the  recoil,  becomes 

?OR  PNEUMATIC  OR  HYDRO  PKBUMATIC  3Y3TIM8; 
If 

paj  -  initial  air  pressure.   (Ibs/sq.in) 
paf  =  final  air  pressure     (Ibs/sq.in) 

Paf 

— —  =  m  =»  ratio  of  compression 

Pai 

VQ  »  initial  air  volume 

Vf  *  final  air  volume 

Ky  *  recuperator  reaction 


383 


Un 


T\\\\\\V?      \\\\\\\\ 


Z1C 


K\\\\\\V> 


oo 

ul 


VA'.W 


384 


Vf  k  »f 

/   Pa*  »  -  -  Pai   »5  / 


1-k    V,         V0 


Paf    ?o  k 

Now  m  »  -  3  <.r~)  3  the  ratio  of  compression 
Pai    vf 

Therefore,  the  work  of  compression  becomes  in  terms 
of  m,  and  the  initial  volume, 


I»(p)  -  1]   ft.  Ibs.     (1) 
where  pas  is  in  Ibs.  per  sq.  ft.  and  VQ  in  cu.  ft. 


[.(,  -u    ft.  lba.  (1., 

12   k-1       k 

when  VQ  is  in  cu.  inches  and  pa^  in  Ibs.  per  sq.  in. 
The  recuperator  reaction,  becomes  for  any  displacement 
X  in  the  recoil, 


Pa  *a  «  Pai 


V0   k 


-A  . 

o  flax 


where  pai  is  in  Ibs.  per  sq.  in.  and  Aa  in  sq.  in. 
Jf  in  inches  and  VQ  in  cu.  inches.   The  initial  volur 
becomes, 

v0  •  V  ^7- (s) 

where  b  =  length  of  recoil.   With  the  oil  in  direct 
contact  with  the  air,  we  will  assume  that  the 


385 


temperature  remains  approximately  constant  through- 
out the  recoil  and  k  Mill  be  taken  at  1.1 

With  a  floating  piston  interposed  between  the 
oil  and  air,  or  with  a  pneumatic  recoil  system,  we 
will  assume  a  negligible  radiation,  that  is  the  com- 
pression approaching  an  adiabatic  condition. 

Hence  k  will  be  assumed  =  1.3 

FOR  SPBIH3  RITUHH  SYSTEMS: 


If 

So  *  initial  spring  recuperator  reaction 
Sf  *  final  spring  recuperator  reaction 

Then  the  potential  energy  stored  in  the  spring  at  the 
end  of  recoil,  becomes 

b  Sf-Sft 

P,g,   .  /      (*     +  _L_2.x)<ix. 


•<VSf>f  (4) 

b" 
and  if  b  is  inches,  we  have  P,E,*(So+Sf )—- 

mm 

The  reaction  exerted  by  the  springs  at  any  displace- 
ment of  the  recoil  X,  becomes 

sf-s0 

Kv  *  So  +  — r    x  (5 ) 


RECOIL  BRAKES.         In  the  broad  classification 
of  recoil  brake  systems,  we  have 
those:  (a)  where  the  brake  system 
is  independent  of  the  recuperator 
system,  (b)  where  the  brake  system 

is  part  of  or  connects  with  the  recuperator  system. 

(a)     In  consideration  of  independent 
brake  systems,  we  nave  a  further 


386 


classification- 
CD     Brakes  with  throttling  orifice 
through  the  recoil  piston,  the  vary- 
ing aperture  during  the  recoil  being 
produced  by  the  difference  in  areas  of  the 
constant  apertures  in  the  piston  and  the 
area  of  the  bar  or  rod  of  varying  depth 
or  diameter  fixed  to  the  recoil  cylinder 
and  moving  through  the  aperture;  or  the 
throttling  nay  be  around  the  piston  by 
varying  grooves  in  the  cylinder  walls 
along  the  cylinder. 

(2)  Brakes  with  varying  apertures  through 
the  recoil  piston,  the  aperture  being 
cut  off  during  the  recoil  by  a  rotating 
disk  about  the  axis  of  the  piston,  the 
disk  being  rotated  during  the  recoil  by  a 
projecting  "toe"  engaging  in  a  helicoidal 
groove  in  the  cylinder  wall.  This  form 
of  brake  is  known  as  the  Krupp  valve  and 
is  extensively  used  not  only  by  the 
Krupp  but  other  types  as  well. 

(3)  Brakes  with  the  throttling  taking 
place  around  the  piston, [not  through 
as  in  (1)  and  (2)],  through  a  sleeve 
perforated  with  boles  along  the  recoil. 
The  piston  cuts  off  the  number  of  boles 
during  the  recoil  thus  decreasing  the 

effective  throttling  area. 

(4)  Brakes  with  the  throttling  taking 
place  through  a  spring  controlled  valve. 
With  independent  brake  systems  the  spring 
valve  is  contained  in  the  piston.  Where 
the  brake  is  part  of  the  recuperator  the 
throttling  takes  place  through  a  fixed 
orifice  sonewhere  between  the  two  cylinders 

(5)  Brakes  with  a  constant  orifice.   On- 


387 


less  the  air  pressure  is  fairly  large, 
and  the  throttling  relatively  small 
constant  orifice  control  should  be  avoided 
since  it  gives  a  large  peak  in  the 
braking. 

In  consideration  of  brake  systems  as  a  part  of  the 
recuperator,  the  throttling  takes  place  between  an  orifice 
fixed  somewhere  between  the  two  cylinders,  and  usually 
of  the  spring  controlled  type  though  sometimes  with 
high  air  pressures  a  constant  orifice  may  be  used. 

In  general  it  nay  be  stated  when  the  recoil  energy 
is  large  the  throttling  may  be  very  satisfactorily  con- 
trolled, as  in  brake  systems  of  the  type  (1),(2)  and 
(3).   Where  the  energy  of  recoil  is  small  as  in  small 
caliber  guns,  the  throttling  areas  especially  at  the 
end  of  recoil  must  be  small.   This  can  not  be  satisfact- 
orily met  with  "bars"  or  "grooves"  due  to  the  inherent 
tolerance  making  very  often  the  clearance  greater 
than  the  required  throttling  areas  towards  the  end  of 
recoil.   This  difficulty  has  been  repeatedly  met  in  the 
design  of  small  recoil  systems.   On  the  other  hand 
spring  controlled  valves  are  admirably  adopted  for 
small  recoil  systems,  since  the  throttling  towards  the 
end  of  recoil  can  be  finely  controlled. 

COUNTER  RECOIL  SYSTEMS  OR  HUNHIKJ  70BWARB  BRAKES: 

In  the  classifications  of  counter  recoil  systems, 
we  have  two  general  types  of  systems: 

(1)  Where  the  brake  comes  into  action 
daring  the  latter  part  of  the  counter 
recoil. 

(2)  Where  the  brake  is  effective 
throughout  the  counter  recoil. 

With  (1),  the  buffer  action  can  only  take  place 
after  a  displacement  of  the  void  (the  displacement 


388 


of  tha  recoil  piston  rod  »  Ar  «  b),  which  with  guns 
of  large  piston  rods  May  be  a  considerable  part  of 
the  counter  recoil. 

With  (2)  the  buffer  must  be  filled  daring  the 
recoil,  otherwise  no  buffer  or  braking  action  can 
take  place. 

The  brake  with  systems  where  the  buffer  action 
takes  place  towards  the  end  of  counter  recoil,  con- 
sists usually  of  a  buffer  chamber  as  an  extension  of  the 
recoil  cylinder  in  the  front  and  spear  buffer  pro- 
jecting from  the  front  side  of  the  piston  or  with  a 
buffer  chamber  within  the  piston  rod  itself  the  spear 
buffer  rod  being  attached  to  the  front  bead  of  the 
cylinder.   In  the  former  type  we  must  have  a  projectory 
chamber  from  the  cylinder,  while  in  the  latter  we  must 
have  a  larger  piston  rod  with  consequent  larger  void 
to  overcome  during  the  counter  recoil. 

With  guns  of  high  elevation  in  order  to  meet 
the  demands  of  counter  recoil  at  maximum  elevation, 
we  have  a  surplus  potential  recuperator  energy  in 
the  recuperator  and  no  means  of  checking  or  regulating 
the  velocity  during  the  greater  part  of  horizontal  recoil: 
therefore  at  the  initial  condition  of  counter  recoil 
stability,  we  have  unfortunately  an  inherent  con- 
dition of  a  large  buffer  force,  making  the  mount  un- 
stable at  the  end  of  counter  recoil. 

Therefore,  this  type  of  counter  recoil  brake, 

which  is  effective  only  during  the  latter  part  of 
counter  recoil  should  only  be  used  for  guns  of  low  ele- 
vation. 


Counter  recoil  brakes  of  type  (2)  which  fill 
during  the  recoil  end  are  effective  throughout  the 
counter  recoil,  are  really  the  standard  form  of 
counter  recoil  regulator  to  meet  the  varying  con- 
ditions required  in  modern  artillery.   Varying  forms 
of  this  type  of  brake  are  used.  Thus  in  the  Filloux 
and  Schneider  reeoil  system  the  buffer  is  at  the  end 


389 


of  a  counter  recoil  rod  which  serves  also  as  a 
throttling  bar  through  the  recoil  piston.  The 
buffer  head  is  enclosed  and  slides  within  a  buffer 
chamber  in  the  piston  rod.   Apertures  near  the  piston 
in  the  piston  rod  adait  the  oil  daring  the  recoil  into 
the  buffer  chanber,  the  oil  passing  through  a  valve 
in  the  buffer  bead  which  can  open  during  the  recoil 
and  closes  during  the  counter  recoil  as  in  the 
Schneider  material.   In  the  Filloux,  though  we  have 
a  filling  in  buffer  in  the  recoil  piston  rod,  the  buff- 
ing action  takes  place  only  at  the  end  of  counter  re- 
coil but  a  positive  filling  is  ensured.  The  velocity 
of  counter  recoil  is  maintained  low  in  this  system 
by  lowering  the  recuperator  pressure  during  the 
greater  part  of  counter  recoil  by  throttling  through 
a  constant  orifice  in  tbe  air  cylinder. 

Various  forms  of  filling  in  buffers  are  shown 
in  figs. (1), (2), (4). 

APPROXIMATE  FORMULA  FOR  If  tbe  total  resistance 
TOTAL  RESISTANCE  TO  to  recoil  is  assumed  con- 
RECOIL.  stant  throughout  the  re- 

coil, we  have  when  the 
recoil  is  along  the  axis 
of  the  bore,  which  usually  occurs  in  practice,  that 

t    ..a 
7  "r  Vf 


b-B+?fT 
where 

B  *  free  reeoil  displacement  during  powder 

period. 
T  *  tine  of  powder  period. 

wv  +  4700  w 
V  «  •          *  max.  free  velocity  of  recoil. 


"r  (ft/sec) 

b  *  length  of  recoil,  (ft) 

Now 


B  *  Ct  VfT  and  T  *  Ct  — 


390 


where 

uo  =  travel  up  the  bore  and  vo  *  muzzle  velocity. 

Substituting, 


f 
v        v 


—  uo  *  c  ~~  uo 


This  value  of  E  may  be  obtained  in  another  way, 

• 

"T     v, 

B  =  C(  -  )u0  -  C  —  u 
r 

hence 


0 
" 


1 

K  »  7  mr  vf< * > 

vf        °o 

b-C  C  —  u0+C  Vf — 

vo       vo 


-  7  -r  vf< 


uovf 

b*(C,-C  C  )  - 
vo 

Mr.  C.  Bethel  found  from  computation  on  a  series 
of  guns  of  various  calibers  that  the  value  (Cj-C1C  ) 
eould  be  represented  very  closely  to  a  linear  function 
of  the  diameter  of  the  bore,  that  is 


Cf-CtCf 


where 

d  =  diameter  of  the  bore,  (in) 
If 

uo»  travel  up  the  bore  (in) 

?o=  nuzzle  velocity  (ft/sec) 

Vf  »  velocity  of  free  recoil  (ft/sec) 

b  *  length  of  recoil  (ft) 

then 

Cf-  CtCt  »  .096  +  .  0003  d 


391 


and  we  bare,  K 


2  uovf 


[b+(.096+.0003  d) 


(Bethels  Formula) 

The  formula  applies  to  a  constant  resistance  to  recoil 
and  is  especially  useful,  since  the  computation  of  E 
and  T  are  not  needed. 

GENERAL  EQUATIONS  OP     The  characteristics  and 
RECOIL  AND  COUNTER    functioning  of  the  various  re- 
RECOIL.-  RECOIL       coil  systems  may  be  shown  in 
SYSTEMS.  an  unique  way  by  a  study  of  the 

equations  of  recoil  and  counter 
recoil.   Let  K  =  the  total  resistance  to  recoil  assumed 

constant  throughout  the  recoil,  (in  Ibs) 

Pb  =  powder  pressure  on  breech 

p  *  the  pressure  in  the  recoil  brake  cylinder. 
(Ibs/sq.in) 

A  =  the  effective  area  of  the  brake  piston.  (sq.  in) 
py  «  the  recuperator  pressure  (Ibs/sq.in) 

AT  »  the  effective  area  of  the  recuperator  pis- 

ton (sq.in) 

B  =  pA  +  Pv*v  s  tne  total  braking,  (in  Ibs) 
R_  *  the  total  packing  frictions,  (in  Ibs) 
Rg  *  the  total  guide  friction  (in  Ibs) 

R  *  Rp+Rg»the  total  recoil  friction  (in  Ibs) 

0  »  angle  of  elevation  of  the  gun. 

X  =  displacement  from  battery  along  the  recoil 

(in  ft) 

b  =  total  length  of  recoil  (ft) 
Then  during  the  recoil 

dv 
Pjj-K  *  mr  —  during  the  acceleration 

-  K  -  m_v  T-  during  the  retardation. 

*  at 


392 


The  external  force  on  the  mount  during  the 
acceleration  is 

dv 

Pjj  -  «r  —  *  K,  as  well  as  the  weight  of  the 

dt      recoiling  parts,  and  a  couple 

P^d^,  where  d^  =  distance  from 
the  center  of  gravity  of  recoiling  parts  to  the 

*   *U     V 

axis  of  the  bore. 

During  the  retardation,  the  external  force  on  the 
mount  in  the  duration  of  recoil  is, 

dv 
—  »pv  —  *  K  ,  together  with  the  recoiling 

dx       weight, 

(1)     when  the  brake  system  is  independent 

of  the  recuperator  system,  then 
K  »  B  *  R  -  wr  sin  0 

»  pA+pvAv+R-wrsin  18 
Now  the  hydraulic  pull  becomes, 

C  v* 
pA  -  

•5 

where 

v  *  the  velocity  of  recoil  at  displacement  x  (ft/sec) 

Nx  -  the  throttling  are  at  displacement  x. 

Further,  with  pneumatic  or  hydropneumatic  re- 
cuperators, 

k 
PVAV  '  Pvi 


)  where  k  =  1.1  to  1.3 


and  with  spring  return  recuperators 
PTAT  -  S  -  S0  *  -~— -  x 

Hence  the  total  resistance  to  recoil,  becomes,  with 
pneumatic  recuperators, 

V, 


393 


and  with  spring  return  recuperators 

*        <?— ^ 
cv         af  so 
K  -  —  +  (S0  +  — £ —  x)+R-Wrsin  0 

Thus  we  have  four  components  in  the  total  resistance 
to  recoil, 

(a)  The  hydraulic  throttling  component 
which  varies  as  the  square  of  the 
velocity, 

(b )  The  elastic  reaction  which  in- 
creases as  a  function  of  the  dis- 
placement. 

(c)    The  friction  component  which  for 
convenience  may  for  a  first  ap- 
proximation be  assumed  constant, 
(d)     The  weight  component  which  exists 
when  the  gun  is  elevated  and  is 
subtrative  since  the  weight  com- 
ponent acts  opposite  to  the  brak- 
ing forces. 

(2)     With  a  recoil  system  where  the  brake 
system  is  part  of   or  connects  with  the  recuperator 
system,  we  have  K  *  pi+R-Wrsin  J0,  where  p  is  the  pres- 
sure in  the  recoil  cylinder.   Now,  due  to  the  throttling 
through  the  orifice  valve  between  the  two  cylinders,  we 
have         i  a 

I     C  V 

P  ""Pa  a       where  pa  »  the  recuperator  pres- 
sure on  the  oil  side  of 
the  recuperator  cylinder. 

v  *  the  velocity  of  recoil,  (ft/sec) 

wx  =  the  opening  of  the  orifice- at  the  recoil  x. 
Further,  the  pressure  in  the  recuperator  at  recoil 
x,  in  terms  of  the  initial  pressure  pai«,  becomes, 

,   (  V0   k  i  3 


o  x 
Hence  substituting  in  the  recoil  equation, 


394 


A)A+R-Wr  sin  0 


sin 


which  is  of  identical  form  as  the  equation  for  re- 
sistance to  recoil,  where  the  recuperator  system  is 
independent  of  the  braking  system. 

Thus  again,  we  may  consider  this  recoil  system 
as  having  the  total  resistance  to  recoil  divided 
into,  the  hydraulic  throttling,  the  elastic,  the 
functional  and  the  weight  components. 

It  is,  however,  often  more  convenient  to  con- 
sider the  total  resistance  as  divided  into  "pressure 
drops".   In  considering  pressure  drops  we  refer  the 
pressure  intensities  to  the  effective  area  of  the 
recoil  piston  and  thus  the  friction  and  weight  component 
drop,  becomes, 


R-lfr  sin 


(Ibs  per  sq.in) 


If  8f  *  the  floating  piston  friction  and  Aa  the 
area  of  the  recuperator,  the  drop  across  the  float- 
ing piston  becomes,          g 

pa  -  pa  »  -—  (Ibs.  per  sq.in) 
*a 

Therefore  the  total  resistance  to  recoil  in  terms  of 
pressure  drops,  becomes 


T"  p 


R-Wr  sin  0 


c  v 


,     , 
Pa)  *<Pa  ~  Pa>+Pa 

2    Rf   R-Wr  sin  0 


R-wr  sin  0 


*      A 


395 


STABILITY  COH8IP1HATIQM 

Now  if 

Kh  •  horizontal  resistance  to  recoil 

h  =  height  of  center  of  gravity  of  recoiling  parts 

above  the  ground. 
wg  >  weight  of  the  total 

ls  *  horizontal  distance  from  spade  to  center  of 

gravity  of  W3  with  recoiling  parts  in  battery. 

wc  *  weight  of  carriage  proper  (not  including  re- 
coiling parts) 

lc  »  horizontal  distance  from  spade  to  center  of 
gravity  of  carriage  proper. 

Vr  *  weight  of  recoiling  parts. 

lr  *  horizontal  distance  from  spade  to  center  of 
gravity  of  recoiling  parts  in  battery. 

e  »  constant  of  stability 

then  since  Wslg«  WrlP  +  *clc  for  any  displacement  x, 
the  stabilising  moment  becomes,  Wr(lr-  x  cos  0)+Wclc=Wslg 
-  Wrx  .  Therefore,  with  a  given  Margin  of  stability, 
we  have,  KDh  *  c(Wsls  -  Wrx)  and  hence  for  a  constant 
margin  of  stability  throughout  the  recoil  at  horizontal 

elevation, 

e  W81,   e  *r 

b       b 
That  is,  the  resistance  tp  recoil  at  horizontal 

recoil,  should  decrease  with  the  recoil  consistent  with 
this  equation. 

When  a  constant  resistance  is  maintained  through- 
out recoil  at  horizontal  elevation,  Kh  should  be  limited 
consistent  with  stability  in  the  out  of  battery  position. 
Advantage  of  the  total  resistance  to  recoil  following 
the  stability  slope: 

(1^     More  energy  is  dissipated  by  the 
brake  during  the  powder  period,  by  fol- 
lowing the  stability  slope  and  thus  gives 
a  greater  decrease  of  the  recoil  dis- 
placement during  the  powder  period. 


396 


(2)     The  braking  forces  being  bigber  during 
the  greater  part  of  the  recoil,  the  re- 
maining energy  or  energy  of  constrained 
recoil,  is  dissipated  in  a  shorter  re- 
coil displacement. 

Hence  the  total  recoil  displacement  is  decreased 
over  that  with  a  constant  resistance  to  recoil. 

Farther,  since  the  stability  slope  causes  a 
smaller  resistance  to  recoil  in  the  out  of  battery 
position  with  a  longer  recoil,  the  total  resistance 
to  recoil  if  Maintained  constant  throughout  recoil  Bust 
be  smaller,  and  the  recoil  displacement  greater  for 
a  given  energy  than  when  the  resistance  to  recoil  fol- 
lows tba  stability  slope. 

The  relation  can  be  shown  analytically  as  follows: 
Assuming  a  constant  resistance  to  recoil  maintained 
during  the  powder  period  and  varying  with  the  stability 
throughout  the  remaining  part  of  the  recoil,  we  have 
for  a  variable  resistance  to  recoil  throughout  recoil, 
K0T  K0T* 

Vr  '  Vf  '  V;   '«•  '  B  '  T~ 

•r  "r 

where  KQ  =  the  resistance  to  recoil  maintained  constant 
during  the  powder  period. 
Since      c  Bgl    c 


the  stability  slope  becomes,  m  »  -  ( 


therefore,  the  resistance  to  recoil  in  the  oat  of 
battery  position  becomes,  k  *  KO  -  m(b-Sr),  we  have 

therefore, 


r 


Substituting  for  Vp  and  Er,  we  have  solving  for  b, 


b,  »  B  *    (1-  f    )±     *o  (1-  f 

&   in—      m  6    T* 

where  A  »  -  mr  ?j  »  energy  of  free  recoil 


397 


o 


398 


c.  *r 


stability  slope 


K0 


With  a  constant  resistance  throughout  the  recoil, 
K(b-Br)  -  ^  mrV«    (1) 

KT*  KT 

where  E_  »  E  -  - —  and  V_  *  V*  -  — - 

2mr  mr 

c  W.l_  c  W» 
and  K  »  — r8— * — E  b    (2) 

a       h 

Combining  (1)  and  (2)  we  obtain  the  length  of 

recoil  for  a  constant  resistance  throughout  recoil, 
and  consistent  with  the  out  of  battery  stability, 

ca   VfT-E    i   / 

bc  *  T  -(— T— >±  r-  /[•(VfT-EJ-C.j'^BlA+C^B-vyr)] 

£t  m         o         o  m 

where      r  w  l       r  n 

3   S  v*TfB 

C-  *  — ;   m  »  stability  slope 

h          h 

A  *  -  nrVf   energy  of  free  recoil. 

bv   Length  of  Recoil  for  Variable  Resistance 
The  ratio  r—  *  — — — — — — — — — — —————— 

bc   Length  of  recoil  for  constant  Resistance 

to  Recoil 
to  Recoil 

gives  the  percentage  of  recoil  by  following  the 
stability  slope  to  that  of  a  constant  recoil  consistent 
with  stability  in  the  out  of  battery  position. 

The  relation  can  be  shown  graphically,  fig.C   ). 
The  ordinates  to  the  line  A8  represent  the  maximum 
possible  overturning  force  consistent  with  stability. 
The  slope  of 


399 


cWj,  C  Wglg 

AB  «  — —   and  the  ordinate  oA  »  — — —  ,  Main- 
fa  h 

taining  a  constant  resistance  to  recoil  during  the 
powder  period  consistent  with  stability  we  have 
ordinates  to  DE,  in  the  powder  displacement  oH.  The 
resistance  to  recoil  decreases  according  to  the  EF  con- 
sistent with  a  constant  margin  of  the  stability. The  area 
OD,   EF  Go,  represents  roughly  the  energy  of  recoil 
A  »  -  mr  Vf  •   If  now  a  constant  resistance  is  to  be 
maintained  we  have  diagram  o  J  P  C  where  the  constant 
resistance  to  recoil  o  J  =  C  P,  and  CP  =  c  8  C,  that 
is,  is  consistent  with  a  given  margin  of  stability 
in  the  out  of  battery  position,  and  further  the  area 
oJPC»A*-jmrVf  (the  energy  of  free  recoil). 

METHODS  OP  CALCULATING     With  independent  recuperator 
THROTTLING  ORIFICES.    systems,  the  throttling  is 

usually  either  by  throttling 
grooves  or  bars  or  by  a  mechan- 
ically controlled  orifice  as 

in  the  Krupp  valve  mechanism  previously  described. 
Let  us  now  consider  the  necessary  throttling 
orifice  variation  along  the  recoil. 

Daring  the  powder  period,  we  have  two  methods, 

(1)  To  maintain  by  a  proper  variation 
of  the  throttling  grooves  a  constant 
resistance  to  recoil  during  this 
period. 

(2)  To  maintain  a  constant  orifice  during 
the  powder  period. 

In  method  (1),  we  have, 

r( )   +  Rt-  Wr  sin  £i  »  K    A  con- 

Vo~AX  stant 

during 

the  powder  period.   Therefore 

C  A-  V 


=  13.2  /  K  -  pa  -  Ft  +  Wr  sin  0 


400 


where  K  »  —  —  —   for  a  constant  resistance  to  reooil. 
b-E+7fT 


K  »  — ^^— — — — — — —    for  a  variable  re- 

2[b-B+VfT-  5-  — (b-B)J        'Stance  to  recoil. 

tr          2   B- 

f   Vp   .\t 

Pa  *  PaiAv^      '  «  paiAv  approx.  during  powder 
YQ— AX. 

period  unless  the  recoil 
is  relatively  short. 

Rt  >  Rg+  Rp  «  total  friction:  guide  friction  *  pack- 
ing friction, 
further  from  interior  ballistics,  av.  total  powder 

pressure      w  v* 
o 
Pe  a 

2g  a 

w  *  weight  of  shell 

TO  *  »u8«le  vel.  (ft/sec.) 

u  *  total  travel  up  bore  (ft) 

27  a 
Initial  pressure  on  breech  in  gas  expansion  po^  =  —  c 

4 

1.12  pM  (Ibs)     where  pa  »  total  powder 

(b*u)"  pres.ure.(lbs) 

and 

27  P«  / 27   P« 

e  «   u(±I D  t  /i  -  *L  Ji).  -  ! 

16  pe  16  pe 

3     u  wv0+4700  w 

t0  "  —  7-     approx.          Vf  -    —- 

*     ?o  "r 


rfo)   Wp  (W*0.5   w)yQ 

*•»  ~~^r  T    Vf°"~r 


r 


T  *  t  +  t   «  total  ponder  period  (sec) 


401 


and  g  =  Xf0  +  f^  +  total  free  recoil  daring  powder 

period,  (ft) 

Three  points  are  sufficient  for  the  orifice 
curve  during  the  powder  period  and  the  corresponding 
constrained  velocity  and  displacement  to  sub-stitute 
in  the  orifice  equation  with  its  lay  out  are: 

K  lo 
7f  *  Vfo (ft/sec) 


"r 

lii 

2m, 


(ft) 


when  the  shot 

leaves  the  muz- 
zle. 


Ktm 
=  v   --  (ft/sec) 


fm 


where 


'f.*:* 


(ft) 


>   the  maximum  restrained 
recoil  velocity  and 
corresponding  orifice. 


4.r(frvgo) 


T 


+  P^bd 
•r 

K(T-t0) 

Pob 


6«r(V£-70) 


sec. 


Rt 

-  —  (ft/sec) 


<ft> 


At  the  end  of  the 
powder  period. 


402 


After  the  powder  period,  that  is  during  the 
retardation  period,  we  have  for  a  constant  resistance 

to  recoil,  simply, 


and  therefore 


CA    o- 


13.2  /K-pa-Rt+Wrsin  0 

which  gives  up  the  required  throttling  with  a  con- 
stant  resistance  to  recoil  during  the  retardation 

period  of  recoil. 

When  the  resistance  to  recoil  is  variable,  we 
have  during  the  retardation  period,  that 

1  mrvj  -[K  -  2  (X+b-2Er)](b-x) 


K-  £(b+X-2Br)](b-x) 




and  there-  /    „, 
fore  .   >^IK--  (b.X-2Br)](b-x) 

CA1  / ~T 

w  ,  ===^=^==^=:_i.—  (sq.in) 

13.2  /K-pa-Rt+Wr  sin  0 

which  gives  the  required  throttling  with  a  variable 
resistance  to  recoil  during  the  retardation 
period. 

With  a  pneumatic  or  hydro  pneumatic  recuperator 
system,  VQ    k 

Pa  "  PaiM»  ,A  y>      where  k  •  1-1  to  1'3 
o  v  V0-  initial 

volume. 


403 

St~  S6 
With  a  spring  return  recuperator,  pa  »  So  »  •  •  •  -  X 

b 
b  -  length  of  recoil  (ft)  where  So  »  initial 

compression  of  the 
springs  (Ibs) 
S  *  final  compression  of  the  springs  (Ibs)  and 


••—  »  2  approx. 
so 

In  method  (2),  with  a  constant  orifice  during 
the  powder  period,  we  have 

cVv1  dv 

*  *  iTTTf  •  p«  -"**"«•  ain  *  '  "r  JT 

Since  an  integration  of  this  equation  is  complicated 
an  approximation  is  made  by  assuming  the  recoiling 
mass  to  recoil  during  the  powder  period  "a"  given  mul- 
tiple distance  of  the  free  recoil  displacement  when  the 
shot  leaves  the  bore.  Let 

Er  *  length  of  constrained  recoil  during  powder 

period,  and  corresponding  length  of  constant 
orifice  (inches) 

u  »  total  travel  of  shot  up  bore  (inches) 
I  *  constant  from  (2  to  2.5)  use  2.24 
w  *  weight  of  powder  charge  (Ibs) 
W  s  weight  of  projectile  (Ibs) 
Pn  *  total  hydraulic  pull  (Ibs) 
wx  =  area  of  orifice  (sq.in)  at  recoil  displace- 
ment x  (in) 
A  »  effective  arc  of  hydraulic  recoil  piston 

(sq.in) 

c  *  coefficient  of  contraction  -  -  -  0.85  to  0.9 
d«  -  S.  G.  of  fluid  »  0.849 

«  a     2   u  (1) 

Now  the  total  drop  in  pressure  through  the  recoil 
orifice,  becomes, 


404 


7(d0  62.5)A*  7X 
p  »  — — — — —     (ibs  per  sq.ft)(See  Hy- 

gc  wx  dro  dynaaics) 

or 

62.5  d0  A»  7» 

P  *  —    (Ibs.  per  sq.in) 

64.4  x  144  c'wj 

During  the  retardation  period  of  the  recoil,  we 
have  fro*  the  equation  of  energy, 

K(b-x)   »   "r  .  »   64.4  K(b-x) 

— —  *  -  7-      hence  7-  »  ,.      ' 

12      *   «  12     »r 


therefore,  .*,,/u  \ 

62.5  dA  K(b-x) 


12  x  144  c'wj  Wr 


and 


. 
w.  -.0361  -^  -  (2) 


d0A8K(b-Br) 
.0361  i*J-  -  -  —  (3) 


which  gives  the  orifice  at  any  displacement  x  in  terms 
of  the  total  resistance  to  recoil,  R  and  the  total 
hydraulic  pull  Ph. 

When  the  resistance  to  recoil  is  made  to  conform 
with  the  stability  slope,  we  have 

t    64.4[K-0.5»(b+X-2Er)](b-X) 
12  wp 

62.5  d0     A*[K-0.5B(b+X-2Er)](b-X) 
P 


12  x  144  c*  w*  Wr 


405 

hence  w.  *  .0361  • — — — — 

C*  "r  Ph 
and 

d0A*[K-0.5»(b+X-2Br>](b-Er) 


.0361 


C"  Wr 


Further 

P  -  K  +  W  .in  *  -  R  - 


for  pneumatic  or  hydro  pneumatic  recuperators, 
and  ss 

Pb  -  8  +  fr  sin  0  -  8t  -  (S0  +  — ^  x  ) 


for  spring  return  recuperators. 

METHODS  OP  THROTTLING     (1)     The  simplest  net  hod 

•f  throttling  is  by  vary- 
ing an  orifice  through  the 
piston  by  throttling  bars 

fixed  to  the  recoil  cylinder 

and  moving  in  the  apertures  through  the , piston.  Let 
wx  *  the  throttling  area  (sq.ia)  as  previously 

calculated. 
S  *  width  of  throttling  bar  or  whole  in  piston 

(inobes) 
b  -  depth  of  hole  in  piston  fro*  cylinder  surface 

(in) 

d  =  depth  of  throttling  bar  (inches) 
d0  *  initial  of  bar  (inches) 
n  »  number  of  bars  (usually  n  =  2 
Then  the  initial  or  maximum  opening 

w0  »  nX(b-d0)(sq.in)  approx.  and  for  any  other 
point  in  the  recoil, 

wx  *  nS(h-d)  (sq.in)  approx, 
With  grooves  in  tbe  cylinder  wall. 

wx  *  n  S  d  (sq.in)  where  d  »  depth  of  rectangular 

groove  (in) 


406 


(2)  When  the  throttling  takes  place  around 
a  long  buffer  rod  of  varying  diameter 

and  passing  through  a  circular  hole  in  the 
piston,  as  in  the  Schneider  material, 

we  have,  if 

0  «  diam.  of  bole  in  cylinder  (sq.in) 
dx»  diam.  of  buffer  rod  passing  through  hole  in 

cylinder  (sq.in) 
then 

»x  -  0.7854(D*-dx)  which  gives  the  variation  of 

the  diameter  of  the  buffer 

rod  along  the  recoil.   la  the  Pilloux  recoil  mechanism, 
we  find  grooves  of  varying  depth  in  the  buffer  rod, 
engaging  with  holes  through  the  piston.  The  object 
of  this  arrangement  is  to  pass  from  one  set  of  grooves 
to  another  by  turning  the  buffer  rod  on  elevating  the 
gun,  thus  making  it  possible  to  shorten  the  recoil 
on  the  elevating  the  gun. 

If  n  *  number  of  grooves  engaged  during  the 

recoil, 
s  =  width  of  groove  (in)  and  d  =  depth  of 

groove  (in) 
then  wx  =  n  3  d. 

(3)  When  a  constant  orifice  is  main- 
tained during  the  powder  period  we  may 
use  the  so  called  Krupp  valve,  which  has 
bad  a  wide  application  for  artillery 
brakes. 

The  initial  orifice  is  closed  uniformly  by  a  disk 
on  the  piston  rotated  by  a  heliooidal  groove  in  the 
cylinder  wall  of  constant  pitch.   Let 

00  *  initial  angle  moved  by  valve  disk  during 

powder  period  before  engaging  the  throttling 
area  in  the  piston. 
0t  *  angle  moved  by  valve  during  the  retardation 

period. 

p  *  pitch  of  helieoidal  groove  in  cylinder  wall 
(inches)  (  Linear  displacement  per  complete 


407 


408 


revolution  of  disk.) 
ro  *  radial  of  cylinder  (inches) 

r  *  radios  to  bottom  of  tbrotting  opening  con- 
tour (inches)  then  the  number  of  turns  » 

0 

—  and  the  linear  displacement  x,  becomes, 

2n 

0 
x  -  —  p 

2x 

Hence  with  a  constant  pitch  with  the  total  recoil 
displacement  b  inches,  we  have 

3xb 


hence     2n(b-Br) 

0  *   ' 

Further  the  throttling  area  becomes,  dwx  »(— — — )d0 
hence     *   •  .  r» 
wx  -  /  *  -2s d0 

0 


dx  (sq.in) 


For  computation  and  design  it  is  more  convenient 
however,  to  express  the  throttling  area  in  terms  of 
the  displacement  from  the  end  of  recoil,  since  the  area 
is  zero  at  this  point  and  opens  up  gradually  to  its 
maximum  near  the  battery  position.  We  have  then, 


w  »  /(b"x)  Mre~'  >  d  (fc-x)  wnepe  r  ,  o,  where  x  »  b, 


In  the  forn  of  a  summation  which  lends  a  simple 
practical  method  of  laying  out  the  contour  of  the 
aperture  in  the  piston,  we  have,  if  the  displacement 
of  recoil  is  divided  into  "n"  parts  from  Er  to  b, 


409 


Starting  from  the  out  of  battery  position, 
we  bare,   n^-r^) 

*„  *  I A(b-X__r)   (sq.in) 


'n-i 


-Xn.)  (sq.in) 


»a-g  M     " A(b-X_)  (sq.in) 


and 

*g  *  ~  2o(ro~rn-g>A(b~xn-g>  Orifice  area  at 

point  g  from  the 

out  of  battery  position. 


Thus  from  a  step  by  ste{.  process  ire  lay  oat  the 
contour  of  the  aperture  in  the  piston,  and  the  total 
area  of  the  orifiee  at  any  displacement  of  the 
recoil,  measured  from  the  out  of  battery  position, 
must  equal  the  required  throttling  area  at  this 
point. 

(4)     Another  form  of  geometrical 

throttling,  devised  in  order  to  effect 
variable  recoil  consists  essentially 
of  cutting  off  holes  in  a  perforated  sleeve 
by  the  piston,  the  throttling  taking 
place  through  the  sleeve  in  the  front  and 
rear  of  the  piston.  We  have  therefore 
two  distinctive  throttling  drops,  that 
in  front  of  the  piston,  and  to  the  rear 
of  the  piston  through  the  recoil  sleeve. 
If  wx  =  the  throttling  area  in  front  oi  the  piston  at 

any  point  in  the  recoil  (sq.in) 
wy  »  the  throttling  area  to  the  rear  of  the  pis- 
ton at  any  two  points  in  the  recoil  (sq.in) 


410 


px  »  the  drop  of  pressure  through  the  throttling 
areas  wx,  in  the  sleeve,  (Ibs/sq.in) 

Py  *  the  drop  of  pressure  through  the  throttling 

areas  wy,  in  the  sleeve  (Ibs/sq.in) 
We  have  for  the  total  drop  P 


P  -  Px  +  P, 


(assuming  the  throttling 


175  wx    175  Wy     constant  C  the  same) 
henee 


F  *  — — — —  (— —  +  — 

175   «•   w; 


where  *c  is  the  equivalent  throttling  area  and  corres- 
ponds to  the  area  obtained  in  the  previous  throttling 
area  calculations. 
In  general 

"•I  **!**!*    »T 

when  we  have  a  drop  of  pressure  due  to  throttling  through 
various  orifices  in  series. 

With  only  two  throttling  drops,  we  have 

*x  "v 

we  *      *  ~~-~    and  »x+wy  *  constant. 

/  m»  +  ** 
wx   "y 

Prom  these  two  equations,  we  have  at  the  maximum 
value  of  wa, 

"x  "  *y 

Hence  in  laying  out  the  holes  in  a  sleeve  valve, 
we  place  the  piston  at  its  displacement  corresponding 
to  naximum  throttling,  that  is  at  the  point  of  the 


411 


maximum  retarded  or  constrained  Telocity,  making  the 
throttling  drop  on  either  side  equal. 

The  process  of  laying  out  the  required  orifices 
and  corresponding  holes  is  as  follows: 

(a)     At  max.  throttling  displace* 

ment  corresponding  to  max.  retarded 
velocity  in  the  recoil, 

Px  »  P_  «  -  and  wx  *  wy  but  since  we  have  a  void 

in  back  of  the  piston  due  to 
the  displacement  of  the  piston 
rod,  P  =  P  i.e.  the  total  drop 
3  the  pressure  in  the  recoil 
..v  cylinder. 

c  A  7 


13.2 

2 

and         2  C  A  Vxm 


13.2 

(b  )     Next  move  the  piston  from 

the  position  of  max.  velocity, 
a  unit  distance  equal  to  the 

width  of  the  piston  in  the 

direction  of  recoil. 
The  area  to  the  rear  -  w_  = 

"c 

-r  ,  since  no  openings  have  been  uncovered 

m 

in  the  rear. 

The  area  to  the  front  is  obtained  from  the 
equations, 

~  -  —  +  —  where  wce,  wxo  etc.  are 
ct   xi    y   the  throttling  areas  at 
max.  velocity  and  wc^  wx^  etc  are  the  throttling 

areas  at  a  distance  from 

the  position  of  max.  velocity  equal  to  the  first 
unit  displacement,  hence 


412 


(sq.in) 

(c)     Next  move  the  piston  another 

unit  distance  in  the  direction 
of  recoil,  the  area  to  the 

w     w         rear, 
c   f  c 

hence      wc  w 

*x   *       *  (sq.in) 


(d)     Hence  for  all  succeeding  points 

in  the  recoil,  w-g  »  wc  -  wyj- 
and  y& 


"xg  ~ — 

(e)     ID  the  powder  pressure  period, 

we  move  the  piston  backward  towards 
the  battery  position  from  the 
position  of  maximum  velocity 
succeeding  units  to  the  rear  and 
the  process  is  exactly  similar 
as  moving  forward  in  the  direction 
of  recoil. 

THROTTLING  THROUGH  A     With  dependent  recuperator 
SPRING  CONTROLLED     systems,  as  in  the  St.  Cbamond 
VALVE.  recoil  system,  the  drop  of  pres- 

sure between  the  two  cylinders 
(i.  e.  between  the  recoil  brake 
and  recuperator  cylinders*  may  be  obtained  by 
throttling  through  a  spring  controlled  orifice  between 
the  two  cylinders.  A  spring  valve,  however,  may  be  used 
with  an  ordinary  recoil  brake  cylinder,  the  throttling 
taking  place  through  a  spring  controlled  orifice  in 
the  piston. 


413 


Let  p  »  the  pressure  in  the  recoil  cylinder  (Ibs/sq.in) 
a  =  the  area  at  base  of  valve  (sq.in) 
pa  »  pressure  in  receiving  chamber  or  recuperator 

(Ibs/sq.  in  ) 
pai  =  initial  pressure  in  recuperator  (Ibs/sq.in) 

Paf  =  final  pressure  in  recuperator  (Ibs/sq.in) 
Aa  =  effective  area  at  top  of  valve  (sq.in) 

at  *  area  of  valve  stem 

S0  =  initial  spring  compression  (Ibs) 

Sf  =  final  spring  compression  (Ibs) 

A  *  effective  area  recoil  piston 

h  »  lift  of  valve  in  inches 

c  =  effective  circumference  at  valve  opening 
Then,  at  the  maximum  opening,  giving  a  lift  h,  ire 
have  pa-  Pai^a  =  ^f  ^^s^  (approx)  and  when  the 
valve  is  about  closed,  pa-  pafAa  =  SQ  (Ibs  )  (approx) 
When  Aa  =  A,  as  with  valves  in  which  the  spring  is 
entirely  enclosed  in  the  recuperator  chamber,  we  have 
(p—  pa^)a=S£  when  open   (approx)  and  (p-paf)a»S0  when 
closed  (approx.) 

When  the  spring  is  outside  of  the  recuperator 
chamber,  and  a  valve  stem  passes  through  a  stuffing 
box  to  the  outside  of  the  chamber,  we  have 

pa-  PaAa  *  Pa-Pa<a~ai>=(P~Pa>a+Paai  (lbs^ 

Further  at  maximum  opening  of  the  valve  we  have 
for  maximum  throttling 


! 

P-Pai  *   -  T~*      "bere  C0  =  -  to  - 
175   C  h  0.6          0.8 

hence 

C0A  V 
h  =  - 


which  gives  the  lift  of  the  valve  at  max.  opening 
and  corresponding  to  a  spring  reaction  =  Sf  Ibs. 

Therefore  knowing  p,  pai  and  paf  and  solving  for 
the  total  lift  h,  we  have,  for  the  spring  required: 


414 


Initial  load  ................  30   (Ibs) 

Final  load  .................  3f   (Ibs) 

Total  lift  .................  b   (in) 

Spring  constant  ............  Sf"So  (lbs  Per  in> 

b 

which  completely  specifies  the  spring  required  to 

properly  function  the  valve  during  tbe  recoil. 

Now  the  pressure  in  tbe  recoil  cylinder,  is 

K+ffr  sin  t!  Rt 
p  »  -          (Ibs/sq.in) 

A 

vo    * 
and  in  the  recuperator  cylinder,  p_  »  paj  (      ) 

VAX 

(Ibs/sq.in) 

The  maximum  throttling  opening  occurs,  at  dis- 
placement Xm  in  the  recoil,  that  is  at  the  point 
during  the  powder  pressure  period,  where  the  powder 
reaction  just  balances  the  recoil  reaction.  This 
is  slightly  before  tbe  end  of  tbe  ponder  period  and  for 
an  approximation  we  have, 

« 

—  •       where  at  *  2  approx. 


Further  tbe  maximum  constrained  velocity  may  be 

taken  at,  Vr  *  g  7f  where  g  *  0.88  at  short  recoil 

*  0.92  at  long  recoil. 

Therefore  at  maximum  opening  of  tbe  valve  (lift  h") 
we  have, 


K+»  sin  6  -  R         Vo 


and  at   tbe  end  of   recoil, 
sin  ?  -  R 


415 


Now  due  to  the  hydraulic  throttling. 


C0  A  Vr 


/K+Wr  sin  0  -  Rt         V( 
1.2  /  - 


13.2  / pai 


Thus  we  have  a  complete  specification  for  the  design 
of  the  spring.   If  now,  ps  =  Ks+Wr  sin  0  -  Rt  =  pull  at 
short  recoil,  max.  elev.  (Ibs),  ph=Kh~Rt  =  PUH  a*  long 
recoil,  zero  elev.  (Ibs)  Fvj  =  initial  recuperator  re- 
action, required  to  hold  gun  at  max.  elev.  in  battery 
(Ibs),  Fvf»  »  Pyi  *  final  recuperator  reaction  at  the 

end  of  recoil  (Ibs) 

We  have,  at  short  recoil,  max.  elevation,  at  the  be- 
ginning of  recoil,      p      p  . 

T-  a  --  I 

A         A 


Sf  "  T-  a  --   a 


*  —  ;  -  a  +  —  —  a   (Ibs)  with 
A  A    * 

springs 

functioning  outside  recuperator  chamber,  and  at  the 

end  of  racoil,      n 
P 


vi 


° 


(lbs) 


»  — a  -r—  at  (Ibs)  with  springs 

functioning  out- 
side recuperator  chamber. 

The  corresponding  max.  lift  at  short  recoil  becomes, 

3 

Cl     tl  E1 

OA  v  Fvi 


now  Sf-S0  *  —  aa(«-l) 


13.2  c  /  P3~Fvi 
and  the  spring  constant,  Ibs.  per  linear  inch,  becomes, 

Sf-S0   13.2  c  Aa(m-l)Fyi  /  pg-Fvi 

S  =  — 3  - (Ibs. per  in) 

h  i. 

C0A»  V 


416 


From  the  above  equations,  we  see,  therefore,  that 
the  load  on  the  spring  is  large  at  short  recoil  and 

proportional  to  the  difference  of  the  pull  at  max. 
elevation  and  the  initial  recuperator  reaction  and  this 
load  is  increased  proportionally  to  the  valve  stem 
area  and  load  on  the  air.  Therefore  to  decrease  the 
load  on  the  springs,  the  valve  stem  should  be  made 
as  small  as  possible,  only  sufficient  to  carry  the 
spring  load.   The  lift  varies  inversely  as  the  square 
root  of  the  difference  of  the  pull  at  max.  elevation 
and  the  recuperator  reaction,  and  when  this  difference 
is  large  as  in  short  recoil,  the  lift  is  proportionally 
small.   Finally  the  spring  constant  (that  is  the  slope 
of  the  load  -  deflection  chart)  increases  with  the  load 
on  the  air  and  with  the  square  root  of  the  difference 
of  the  max.  pull  and  the  initial  air  recuperator  re- 
action.  On  the  other  hand,  if  the  compression  ratio 
is  low,  approaching  I,  or  if  the  annular  area  or  the 
effective  area  on  top  of  the  valve  is  small,  that 
is,  using  a  large  valve  stem,  we  must  have  a  spring 
of  considerable  deflection  for  a  given  change  in 
load.   When  Aa  »  0,  or  Fyi  »  0,  we  have  no  change  in 
load  in  the  spring  and  the  valve  would  open  a  given 
lift  h,  with  a  corresponding  spring  reaction.   As  the 
gun  recoils,  if  the  lift  and  corresponding  throttling 
area  remained  constant,  the  pressure  would  drop  pro- 
portionally to  the  square  of  the  velocity.  This,  there- 
fore, causes  a  gradual  closing  of  the  valve  since  the 

spring  reaction  must  decrease,  and  we  have  a 
throttling  in  between  an  ideal  spring  controlled  orifice 
and  that  with  a  constant  orifice.  Even  with  this 
arrangement  we  have  a  vast  improvement  over  that  of  a 
constant  orifice  and  the  peak  in  the  throttling  ia 
greatly  reduced. 

Now,  at  long  recoil,  horizontal  elevation,  at 
the  beginning  of  recoil,      p      p  . 

Sf  =  —  a  -  —  Aa  (Ibs) 
A       A 


417 


»  —  —  —  a  (Ibs)     with  spring  functioning  inside 

recuperator  chanber  as  is 
usually  the  case  at  long  re- 

coil (See  St.  Cbaaond  Chapter),  at  the  end  of  recoil, 

Pb     -pv 
S0  =  —  a 

A       A 


bvi 
*  — — —  a   (Ibs)  with  spring  functioning  in- 

A  side  recuperator  chamber. 

Tbe  corresponding  max.  lift  at  long  recoil,  be- 

cones,          3_ 

C0  a"  V 

h  a  ^— — —       (inches) 


13.2  c  /  Pn-*vi 

Further      B  . 
vi 
Sf-S0  =  — —  a  (m-1)  (Ibs)  and  the  spring  constant, 

Ibs.  per  linear  inch,  becomes, 
S*-SA   13.2  c  a(i 


C0  a«  V 

Prom  these  equations  we  see  the  load  on  the 
springs  is  relatively  snail  as  compared  with  short 
recoil,  the  deflection  b  large  and  the  spring  con- 
stant snail. 

Thus,  in  comparing  the  requirements  of  spring 
characteristics  at  short  and  long  recoil  respect- 
ively, we  have, 

(1)  Short  recoil  and  max.  elev.  = 

A  large  spring  reaction  and  small  de- 
flection with  a  spring  constant  having 
a  steep  load  deflection  slope. 

(2)  Long  recoil  and  horizontal  elev.* 

A  small  spring  reaction  and  large  deflection 
with  a  spring  constant  having  a  snail 
load  deflection  slope. 


418 


To  meet  the  requirements  of  (1)  in  the  St.  Chanond 

recoil  system  we  find  Belleville  spring  used;  and  in 
(2)  the  use  of  a  weak  spiral  spring. 

When  a  spring  valve  is  used  without  a  recuperator, 
the  spring  valve  is  usually  located  in  the  piston  of 
the  hydraulic  cylinder.   In  the  design  and  working 
of  this  valve  the  following  points  are  important: 
Let 

Phi  *  the  initial  hydraulic  pull  (Ibs) 

Phf  «  the  final  hydraulic  pull   (Ibs) 

A   *  the  effective  area  of  the  recoil  piston 
(sq.in) 

a   *  the  area  at  the  base  of  valve  (sq.in) 

Pai  =  initial  recuperator  reaction 

Paf  *  final  recuperator  reaction 

Rt  *  total  recoil  friction 
Then  Ph^  +  Paj  +  Rt  -  Hr  sin  0  »  K  at  the  beginning 

of  recoil,  and 

phf  *  paf  *  Rt  ~  wr  sin  ^  *  K  at  the  end  of  recoil, 
hence  Pni  »  K  +  Wr  sin  0  -  Rt  -  Pai  :   Phf»K+Wp  sin 


At  the  beginning  of  recoil, 
p 

—r-  a  (Ibs)  the  pressure  in  the  back  of  the  valve 

being  negligible. 
At  the  end  of  recoil, 


S0  -     a  (Ibs) 

The  throttling  at  the  beginning  of  recoil,  be- 


comes 


a  (Ibs)  and  the  spring 


419 


13.2   a  c  /Pbi(Paf-pai> 


C0  A  *  V 
The  above  equations  show  that  the  maximum  load 

on  the  spring  depends  upon  the  maximum  hydraulic 
load,  the  assembled  load  on  the  minimum  hydraulic  load 
at  the  end  of  recoil,  the  lift  varying  inversely  as 
the  square  root  of  the  maximum  hydraulic  load  and  the 
spring  constant  or  the  compression  deflection  slope 
of  the  spring  being  proportional  to  the  difference 
between  the  final  air  and  initial  recuperator  re- 
action and  the  square  root  of  the  maximum  hydraulic 
reaction. 

The  spring  throttling  valve  has  been  used  success- 
fully with  an  ordinary  hydraulic  recoil  brake 
cylinder,  designed  for  approximately  constant  pull 
throughout  the  total  recoil  as  in  the  lower  brake 
cylinders  of  a  double  recoil  system  or  in  the  brake 
cylinders  of  a  gun  or  sliding  carriage  mount.   Of 
course  it  is  impossible  to  maintain  an  absolute 
constant  braking  resistance  throughout  the  recoil  as 
previously  discussed  but  a  sufficient  approximation 
can  be  obtained  to  justify  its  use. 

In  the  design  of  constant  braking  with  a 
spring  control, we  have  a  spring  valve  seated  in  the 
piston. 

If  the  throttling  takes  place  mainly  through  the 
valve  seat,  we  have  p  a  »  So  *  S  h  where  p  *  pres- 
sure in  the  recoil  cylinder,  (Ibs/sq.in) 

S0  =  initial  spring  load  (assembled  load)(lbs) 

S  =  spring  constant  (Ibs/in) 

a  *  the  effective  area  at  the  base  of  the  valve. 
h  =  lift  of  valve  (inches) 

Now      coA  7         l      l 

h  *  ,   Ca  »  to  

13.2e/p        0.6    0.8 

If  the  valve  is  bevelled  the  throttling  area 
becomes  in  place  of  c  h, 

w  «  *  D  h  sin  0 


420 


whore  D  *  >ean  diai.  of  the  bevel  portion  of  the 

valve  (in) 
tf  »  angle  of  bevel  leasured  with  respeet  to 

the  central  axis  of  the  valve, 
beneo       CQ  A  7  j      1 

h  -  ===-,  C  =  to  

13.2  n  D  sin  0  •  p         0.6    0.8 

To  design  the  spring  we  may  adjust  So  to  give  a  suitable 
value  of  the  spring  constant  S,  by  the  formula, - 

S  > 


RECOIL  THROTTLING  WITH     When  a  buffer  or  regulator 
A  "PILLING  IN*  COUNTER   is  desired  to  act  through- 
RECOIL  BUFFER.          out  the  counter  recoil,  the 

counter  recoil  buffer 
chamber  must  be  filled 
during  the  recoil. 

The  filling  of  the  counter  recoil  buffer  chamber 
during  the  recoil,  affects  the  recoil  throttling  in 
two  ways: 

(1)     The  total  oil  displaced  by  the 
recoil  piston  does  not  pass  through 
the  recoil  throttling  grooves:  a 
part  passing  into  the  buffer  chamber 
in  the  process  of  filling  it  in  the 
recoil. 

(2.)      In  the  buffer  chamber,  we  have  more 
or  less  pressure  during  a  part  of  the 
recoil,  since  if  the  throttling  into 
the  buffer  chamber  is  just  sufficient 
to  fill  during  the  max.  vol.  of  recoil, 
we  will  have  if  the  pressure  in  the 
recoil  cylinder  remains  constant  an 
over  filling  during  the  latter  part 
of  recoil  and  therefore  pressure  in 
the  buffer  chamber,  since  the 
throttling  drop  is  decreased  due  to 
the  decreased  velocity  of  recoil. 


421 


Therefore  tbe  total  hydraulic  reaction 
eo  the  piston  rod  is  somewhat  modified. 
Let  p  *  intensity  of  pressure  in  recoil  cylinder 

(Ibs/sq.in) 

A  *  effective  area  in  recoil  piston  (sq.in) 
Ab  *  effective  area  of  buffer  (sq.in) 
Vx  *  recoil  velocity  (ft.  sec) 
wx  =  recoil  throttling  area  (sq.in) 
ao  -  entrance  throttling  area  for  filling  buffer 

chamber  in  tbe  recoil  (sq.in) 
Pb  *  intensity  of  pressure  in  buffer  chamber 

(Ibs/sq.in) 

Then,  during  the  recoil,  we  have,  for  the  tension  in 
tbe  rod  "Ph" 

Ph  -  p  A  -  pb  Ab        (Ibs)U) 
Tbe  drop  of  pressure  due  to  tbrottling  through 
tbe  filling  in  bole  to  tbe  buffer  chamber,  becomes 

for  continuous   filling, 

_  .1  .«  _» 
co  Ab  vx 

Ph  s  P  ~  Pb  =  -  (Ibs/sq.in)  (2) 
175  a» 

hence  C'*Au  V* 

Ph  =  p(A-Ab)*  °     X    (Ibs)       (3) 
175  a« 


175  a« 

(Ibs/sq.in)  (4) 


A  -  Ab 

Further,  ifith  continuous  filling,  we  have,  for  tbe 
velocity  through  the  recoil  tbrottling  orifice, 

u-V  vx 

YX  „  (ft/sec)(5) 


and  therefore,     ca(A-Ab)*  V* 

P  '  (Ibs/sq.in) 

175  "x  (6) 


422 
Combining  U)  and  (6)  we  have, 


w 


a 
C0(A-Ab)« 


*         /     C'*A>  7* 

13.2  /Pb  -  n  o  i 
175  a« 

which  gives  the  required  recoil  throttling  area 
(assuming  a  density  of  the  liquid  =  53  Ibs.  cu.ft.)  in 
terms  of  the  total  pull  Ph,  the  recoil  constrained 
velocity  Vx  and  the  constant  filling  in  entrance 
area  to  the  buffer  chamber  ao. 

If  the  density  of  the  liquid  is  different  from 
that  of  hydroline  oil  *  53  Ibs/cu.ft.we  have, 

— (sq.in) 

ni*ni?  v* 
288g(Ph  - 


288  g  a« 


C0VX  /D(A  -  Ab)' 

/  • (sq.in) 

12          Cr»  DA»  7« 

-* *-*-) 

288  g  a« 


where  D  »  weight  of  liquid  per  ou.  ft. 

If  we  have  several  contractions  in  the  filling 
in  passage  to  the  buffer  chamber,  we  have  approximate- 
ly assuming  tne  same  contraction  factor  for  the  flow 

°»'   1    1    1  1 

— _  =  —  +  —  +  _  _  _  _  — 

»     a     *  x 


Determination  of  ao: 


If  we  desire  a  continuous  filling  of  the  counter 


423 


recoil  buffer  chamber  daring  the  recoil,  with  a  constant 
entrance  throttling  area  for  filling  the  buffer  chamber, 
we  must  design  ab  for  throttling  at  maximum  velocity  of 
recoil,  since  the  throttling  drop  varies  with  the  square 
of  the  velocity  and  is  a  maximum  at  maximum  velocity, and 
the  pressure  in  the  recoil  cylinder  remains  approximate* 
ly  constant  during  the  recoil. 

If  now,  the  throttling  drop  is  just  equal  to  the 
pressure  in  the  recoil  cylinder  at  maximum  velocity, since 
the  throttling  drop  is  less  at  all  other  velocities  and 
the  pressure  head  the  same,  we  have  a  pressure  in  the 
buffer  chamber  continuously  rising  during  the  latter 
part  of  the  recoil. 

If  the  throttling  drop  at  maximum  velocity  is  less 
than  the  pressure  head  in  the  recoil  cylinder,  we  have 
a  void  in  the  buffer  chamber  daring  the  first  part  of 
recoil  when  the  velocity  of  recoil  is  large,  and  there- 
fore, not  continuous  filling. 

For  continuous  filling,  therefore  pmax  >  pb  at  max. 
vel.  of  recoil  and  therefore  max.  recoil  pressure,  that 

i«       „'*.«  ,,« 

c»  Ab  vmax  C0  AbVmax 

Pmax  i      a,     hence  aQ  >  — —-  ,(sq.in) 

13.2  /  pmax 

which  gives  the  proper  entrance  throttling  area  re- 
quired for  filling  the  buffer  continuously  during  the 
recoil. 

Since,  however,  the  buffer  over  fills  during  the 
greater  part  of  the  remainder  of  recoil,  ao  can  be  made 
smaller  than  required  for  a  continuous  filling  through- 
out  the  recoil  and  yet  have  a  complete  filling  of  the 
buffer  chamber.   In  order  that  the  buffer  chamber  may 
completely  fill,  (though  not  continuously  throughout 
the  recoil)  we  have,  for  the  time  of  recoil,  roughly 


approx. 

PA 


424 


and  assuming  the  pressure  in  the  buffer  chamber  at  any 
tiae  of  the  recoil  snail,       * 

p  =  —  D   (Ibs.per  sq,ft) 
2g 


For  the  filling  of  the  buffer  chamber, 

=  Abb,  where  b  =  length  of  recoil  (ft) 


a0vt 


hence   —  /  ^— >  t  *  Ab  b   and  AQ«CoAbA  b  /-*—  (sq.in) 

CQ  D  *g 

where  b  *  length  of  recoil  (in) 
Ab  =  area  of  buffer  (sq.in) 

Cn  *  contraction  constant  of  orifice  )=  to  

0.6    0.8 

p  =  pressure  in  recoil  cylinder  (Ibs/sq.in) 
A  3  effective  area  of  recoil  piston  (sq.in) 
D  *  density  of  liquid  (los/cu.ft) 

Since   the  pressure  in  the  buffer  is  probably  small  by 
this  method  of  filling,  we  may  neglect  the  total 

buffer  reaction  in  modifying  the  tension  or  pull  in 
the  rod.   Further  the  throttling  in  the  "filling  in" 
buffer,  becomes,     C^AvV8 


b¥x 

approx 


175  a* 


hence  AbVx  *  Qb  constant  -  that  is,  the  flow  into 
the  buffer  may  be  assumed  constant  throughout  the 
recoil,  hence  for  the  main  recoil  throttling,  we 

have>     C*(AV-Qb)* 


175  wj 


C0(A7x-Qb) 

"x  •  ~I 

13.2  /p 


Since,  however,  pfe  (the  pressure  in  the  buffer) 
actually  rises  even  in  this  method   somewhat  towards 
the  end  of  recoil,  Qj,  decreases  with  AVX  and  there— 


425 


fore  by  slightly  modifying  the  true  contraction 
constant  Co,  we  have,      c  AV 

wx  * 

13.2  /~p~ 

which  is  sufficiently  exact  for  ordinary  design. 

For  correct  filling  of  the  buffer  chamber,  the 
filling  throttling  area  to  the  buffer  should  be 
made  variable.  We  nay  plot  this  variable  area 
against  recoil  and  take  its  mean  value  as  an  ap- 
proximation for  the  proper  throttling  area  for 
filling  the  buffer  chamber. 

The  condition  for  ideal  filling  of  the  buffer 
chamber  are,  that  C  Oxax  *  AbVx  and  Pb  »  0  throughout 
the  recoil,  where  ux  =  the  throttling  velocity  into 

the  filling  in  buffer, 

pb  *  the  pressure  in  the  buffer  chamber 
c  =  the  contraction  constant  of  the 

orifice. 
ax  =  the  variable  buffer  filling 

throttling  area  (sq.in) 
By  Bernoulli's  theorem,  we  have, 

DU*  D*x 

p  »  — —  and  p  *  (Ibs/sq.in) 

288g          288g 

where  vx  =  the  velocity  through  the  recoil  throttling 

orifice. 
D  =  the  weight  of  the  fluid  per  cu.  ft. 

A»  " 

* 

and  since  Pn  ~  P  A 


D  Yg 


N  w  —  is  a  variable  in  the  recoil,  and  therefore 

b  the  recoil  throttling  areas  become  modified 
at  any  instant,  such  that, 


426 


_      ~v» 

bcnee  W  ,  _» LJLJL 


A*V*      c 
x       * 

—  A  —  """^ 


288  g  Pn    *  C 


Constructive  difficulties  make  it  impractical 
to  vary  AX  according  to  the  above  theory  in  an 
ordinary  design  but  by  making  ax  =  ao  a  constant, 
and  assuming  p^  small,  we  have  from  the  above  formula, 
that  the  recoil  throttling  area  equals  the  throttling 
area  computed  as  if  no  buffer  existed  in  the  recoil, 
and  lessened  by  a  constant  area    Q 


VARIABLE  RECOIL:-     Stability  consideration:   As  the 
VARYING  THE  RECOIL   gun  elevates  the  overturning 
AS  THE  GUN          moment  decreases,  since  the  per- 
ELEVATES.  pendioular  distance  from  the 

spade  point  or  the  point  where 

the  mount  tends  to  overturn  on  reooil,  to  the  line  of 
action  of  the  total  resistance  to  recoil  decreases  on 

elevation.  Therefore,  since  the  initial  recoil  energy 
is  practically  constant,  it  is  possible  to  decrease 
the  length  of  recoil  considerably  as  the  gun  elevates 
and  yet  maintain  stability.  When  the  line  of  action 
of  the  resistance  to  recoil  passes  through  the  spade 
point,  the  overturning  moment  is  independent  of  the 
magnitude  of  the  recoil  reaction,  and  therefore 
theoretically  the  recoil  can  be  made  as  small  as  the 
strength  of  the  carriage  can  stand. 


427 


Therefore,  the  recoil  limitations  on  elevating 
the  gun  are  clearance  at  maximum  elevation,  as  well 
as  clearance  considerations  at  intermediate  elevations, 
and  the  limitation  imposed  by  stability  for  various 
elevations  of  the  gun. 

The  recoil  may  be  cut  down  in  any  arbitrary 
manner  provided,  that  consideration  be  given  to 
strength,  clearance  and  stability  at  all  angles  of 
elevation.   The  maximum  length  of  short  recoil  depends 
upon  clearance  considerations  at  maximum  elevation, 
while  the  minimum  length  of  long  recoil  depends  upon 
stability  at  horizontal  elevation. 

To  investigate  the  stability  limitations  on  the 
length  of  recoil  at  low  angles  of  elevation,  let 

C  =  constant  of  stability  =  Overturning  moment  , 

Stabilizing  moment 
0.85 

Ar=  initial  recoil  constrained  energy  =  -  »rVr 

(ft/lbs) 
Vr=*  0.9  Vj  restrained  recoil  velocity   (ft/sec) 

w  v  +  w  4700 
Vf  =  =  free  velocity  of  recoil 

*r         (ft/sec) 
u  *  travel  up  bore  (in  ft) 
Er  'displacement  of  gun.  during  powder  period  3 

(w+  0.5  w)u 
2.25  (in  ft) 

*r 

d  =  moment  arm  to  line  of  action  of  total  re- 
sistance to  recoil  (ft) 
b  =  length  of  recoil  (ft) 

Then>  Ar    C0fsls  -  Wr  b  cos  £1) 


b-E, 


and  solving  for  b, 
we  have 


/                                  a                                 dAr 
00)-  *    (Wslg+T*rErcos  0)   -4Wrcos(WalsEr* 


2  Wr  cos  $ 

(ft) 


428 


which  gives  us  the  limiting  recoil  consistent  with 
stability  for  low  angles  of  elevation,  with  a  con- 
stant  resistance  throughout  the  recoil. 

When  the  resistance  to  recoil  is  made  to  con- 
form with  the  stability  slope,  we  have, 

b  A 


s  ~  *r  *  cos  P  )  <*X  *  — 


Solving,  we  have  -EWsla(b-EP) 


Hence,  we  have,  the  quadratic  equation  in  terms  of 
b 

dAp  Wrcos  0   t 

a  ^fr^lg  ft   *  HslsEr  -   —^       BP 

b  — 


wrcos  &  lfp  cos  CT 

Solving  for  b:  we  have, 


Wpcos/) 


/  AP 

EW81S-  /("sis)  ~  2Wrcos  0(—  d+WslsEr- 


which  gives  us  the  limiting  recoil  consistent  with 
stability  for  low  angles  of  elevation,  with  a  variable 

resistance  throughout  the  recoil  conforming  with  the 
stability  slope. 


MITHOD  Of  D1CBKASIHG  THE  LENGTH  OF  RICOIL; 

In  the  layout  design  of  varying  the  recoil  on 
elevation,  it  is  highly  desirable  to  maintain  a  con- 
stant recoil  equal  to  ihat  at  horizontal  recoil  for 
the  first  few  degrees  of  elevation  and  then  begin 
cutting  down  the  length  of  recoil,  to  the  minimum 
recoil  at  max.  elevation,  since  by  this  method  the 
margin  of  stability  increases  as  the  gun  elevates 
and  therefore  exact  stability  at  horizontal  recoil  is 


429 


00 

L 


430 


431 


I 

/ 

\ 

C 

(-—•  -* 

.  —  - 

~  —  • 

•« 

s' 

^ 

$ 

? 

^ 

CJ 

TOM 

JO 

•01 

(D 

V) 

\L 


432 


no  longer  of  vital  consideration  as  horizontal  fire 
in  seldom  used.   In  certain  types  of  recoil  systems 
as  in  the  St.  Chamond  recoil,  the  size  of  the  re- 
cuperator may  be  decreased  by  increasing  the  pull 
at  horizontal  elevation  and  therefore  in  this  type 
of  recoil  it  is  highly  desirable  to  design  to  the 
exact  stability  at  horizontal  recoil,  as  the  gun 
elevates  with  constant  recoil  we  therefore  will  have 
ample  stability  even  at  low  elevations. 

Therefore,  unless  limited  by  clearance  ,  it  is 
desirable  to  maintain  a  constant  recoil  from  0°  to  20° 
elevation,  and  then  cut  down  proportional  to  the 
elevation  to  the  minimum  recoil  length  at  maximum 
elevation. 

MECHANISM  FOR  REDUCING     Variable  recoil  is  obtained 
THE  RECOIL  ON  ELEVATION-  by  decreasing  on  elevation 

the  initial  throttling  areas 
by  turning,  the  counter  re- 
coil buffer  rod  which  contains 

sets  of  the  recoil  throttling  grooves,  as  in  the  Pil- 
loux  recoil  mechanism;  or  by  turning  the  piston  and 
its  rod  with  respect  to  the  rotating  valve,  and  thus 
changing  the  initial  openings  in  the  Krupp  recoil 
mechanism;  or  by  rotating  a  perforated  sleeve  as  in 
the  American  sleeve  valve. 

Two  methods  for  rotating  the  throttling  rod, 
valve  or  sleeve  are  used, 

(1)  by  a  sliding  bar  linkage  as  in  the 
Pilloux  mechanism  or 

(2)  by  a  four  bar  linkage  as  in  the 
Krupp  or  sleeve  valve  recoil  mechanism. 

With  a  sliding  bar  linkage  in  the  elevation  of  the  gun, 
a  cross  head  or  bar  is  moved  in  translation.  The  bar 
contains  a  pin  which  engages  in  a  helical  groove  of 
the  rotating  cylinder,  thus  giving  the  necessary 
rotatory  motion.   With  a  four  bar  linkage  the  valve 


433 


is  turned  directly  in  the  movement  of  the  linkage 
during  the  elevation  of  the  gun. 

(1)     In  a  layout  of  the  sliding  bar 

linkage,  the  distance  of  the  translation 
of  the  bar  or  cross  head  is  fixed  by 
the  pitch  of  the  helix  on  the  rotating 
cylinder  and  the  angle  turned  to  be  turned 
by  the  cylinder.  The  pitch  of  the  helix 
may  not  be  constant  that  is  the  slope 
of  the  helix  may  vary  in  the  revolution. 
With  a  uniform  pitch  or  slope  of  the 
helix,  the  decrease  in  the  length  of  re- 
coil against  elevation  may  not  be 
uniform  but  for  constructive  reasons  it 
may  be  sufficiently  satisfactory. 

Knowing  the  length  of  the  translation  of  the  slide 
we  may  layout  the  valve  mechanism.   In  the  sliding  bar 

linkage  of  the  recoil  mechanism,  the  crank  with 
center  at  the  trunnions  is  made  the  fixed  link,  while 

the  frame  of  the  mechanism  rotates  on  elevation.   If 

now  we  draw  two  circles  with  centers  at  the  trunnions 
and  crank  pin  respectively,  the  relative  displace- 
ment of  the  crosshead  or  bar  is  the  distance  between 
the  intersection  of  these  circles  and  a  line  drawn 
through  the  center  line  of  the  slide  bar.   Constructive- 
ly, it  is  convenient  to  draw  a  secondary  constructive 
circle  tangent  to  the  projectile  center  line  of  the 
initial  position  of  the  slide  bar,  i.  e.  usually  at 
horizontal  elevation.  Then  at  any  elevation  the 
center  line  of  the  slide  bar  must  be  tangent  to  this 
circle.   Hence  the  intersection  of  these  tangents  with 
the  base  circles  of  radii  at  trunmion  and  crank  pin 
respectively  gives  the  relative  displacement  of  the 
slide.  The  proper  position  of  the  crank  pin  with  res- 
pect to  coordinates  with  origin  at  center  of  trunnions 
can  practically  only  be  determined  by  successive  trials 
for  the  proper  movement  of  the  slide  bar. 


434 


(2)     In  a  layout  of  a  four  bar  linkage 
the  angle  of  rotation  of  the  valve 
during  the  elevation  of  the  gun  is  as- 
certained from  the  design  of  the  re- 
coil throttling.   The  gear  turning  the 
valve  may  mesh  with  another  gear  and 
from  the  gear  ratio  and  the  maximum 
turning  of  the  valve  the  angle  turned 
by  the  valve  crank  can  be  determined. 
Knowing  the  angle  turned  by  the  valve  crank  or 
valve  arm  we  nay  then  layout  the  valve  mechanism. 
The  four  bar  linkage  consists  of  the  frame  connecting 
the  trunnion  and  valve  center;  the  fixed  trunnion 
crank  connecting  the  trunnion  and  connecting  rod;  the 

connecting  rod  connecting  the  fixed  trunnion  crank 
and  the  valve  crank  or  arm;  and  finally  the  valve  or 

arm  connecting  the  connecting  rod  with  the  valve 
center.  Tbe  fixed  member  of  the  four  bar  linkage  is 
the'fixed  trunnion  crank"  joining  the  trunnion  to 
the  connecting  rod.   If  ire  draw  two  circles  from 
the  fixed  centers  of  the  trunnion  and  trunnion 
crank  pin  respectively,  the  center  of  the  valve  travels 
along  the  circular  path  with  center  at  the  trunnion, 
while  tbe  crank  pin  of  the  valve  arm  moves  in  a  cir- 
cular path  with  center  at  tbe  fixed  trunnion  crank  pin. 

It  is  important  to  note  that  the  relative  position  of  tbe 
valve  crank  arm  should  be  measured  from  tbe  tangent 
to  the  circle  with  center  at  tbe  trunnions.  Tbe 
relative  angle  turned  by  the  valve  crank  is  therefore 
the  difference  between  the  final  angle  with  respect  to 
tbe  tangent  of  tbe  trunnion  circle  when  at  maximum 
elevation  and  the  initial  angle  with  respect  to  the 
tangent  of  the  trunnion  circle  when  at  minimum, 
usually  horizontal  elevation. 

Constructively,  it  is  convenient  to  draw  a 
secondary  constructive  circle  tangent  to  a 
horizontal  line  through  the  center  of  the  valve  arm. 
Then  the  position  of  the  valve  center  at  any  elevation 
is  the  intersection  of  the  tangent  to  this  secondary 


435 


circle  at  the  given  elevation  with  the  base  trunnion 
circle  of  the  valve. 

If  He  lay  off  from  this  intersection  the  length 
of  valve  arm  to  the  intersection  of  the  trunnion 
crank  pin  base  circle,  we  have  the  position  of  the 
valve  arm  for  this  elevation.   For  the  angle  turned 
we  note  the  angle  made  by  the  valve  arm  with  the 
tangent  to  the  trunnion  base  circle  at  the  valve 
center,  and  the  initial  angle  of  the  valve  am  with 
the  tangent  at  horizontal  elevation.  The  difference 
between  these  angles  is  the  angle  turned  by  the  valve 
arm,  which  multiplied  by  the  gear  ratio  gives  the  actual 
angle  turned  by  the  valve. 

ON  THE  LENGTH  OF  RECOIL  As  before  for  a  grooved 
WITH  A  STATIONARY  SPRING  orifice  we  have  from  the 
CONTROLLED  ORIFICE.  equation  of  energy: 

K(b-x)*  £  mB  v*   (1) 
where  b  =  length  of  recoil  (ft) 

x  =  recoil  displacement  (ft) 

vx=  recoil  velocity  at  displacement  x  (ft/sec) 

mR=  mass  of  recoiling  parts 

and  for  the  total  resistance  to  recoil,  for  a  dependent 
recoil  system  K  =  p  A  +  R  -  Wr  sin  0 

where  p  -  pressure  in  the  recoil  cylinder  (Ibs/sq.ft) 
R  =  total  friction  (Ibs) 

A  =  effective  area  of  recoil  piston   (sq.ft) 

"0*        D  A*VX  D  A3?!   C  V* 

P~Pa  *  -     and  (P~Pa)A*  -  =-    " 
2gc*w«  2gc«w»   "x 


then  since 


,,  N     /0. 
Combining  (1)  and  (2),  wx  = 


+  R  -  »fr  sin  0       (2) 
2KC(b-x)  _ 


m_(K-paA-R+Wrsin  0) 


436 


the  ratio  C  *  — — — ~ — — — —  is  approximately  con- 

K-paA-R+W.sin  0 

stant,  since  the 

variation  of  the  weight  component  Hrsin  0  amd  the 
recuperator  reaction  paA  is  small  compared  with  K. 
Then      2CQ 

wj  =*  (b-x)  where  Co  »  c'c. 

Therefore  the  orifice  variation  is  a  parabolic  function 
of  the  recoil  displacement  and  is  independent  of  the 
initial  velocity  and  therefore  variation  in  the 
ballistics,  and  is  practically  independent  of  the 
•eight  component  and  therefore  of  the  elevation  of 
the  gun. 

In  general,  independent  of  the  method  of  throttling 

the  length  of  recoil  is  practically  independent  of 
variation  in  the  ballistics  of  the  gun  or  in  the 
variation  of  the  elevation  of  the  gun. 

ON  THE  LENGTH  OP  RECOIL     During  the  retardation 

WITH  A  GROOVED  ORIFICE,   period  of  the  recoil,  we 

have,  from  the  equation 
of  energy, - 

K(b-x)=  ;  mr  V* 

where  b  =  length  of  recoil   (ft) 

x  *  recoil  displacement  (ft) 

Vx=  recoil  velocity  at  displacement  x  (ft/sec) 
T.r  «  mass  of  the  recoiling  parts 
K  =  total  resistance  to  recoil  (Ibs) 

hence 

»      2K(D-x). 
V    »  (1) 


HOTB:    Rot  confirmed  by  observed  data.    Bditor. 


437 


D  A'V! 


(Ibs/sq.ft) 


D  A*V* 


p  ,  p  A  ,  »  (2) 

2gC2W«     W« 

Ph  *  total  hydraulic  pull  (Ibs) 
A  »  effective  area  of  recoil  piston  (sq.ft) 
D  »  weight  per  cu.ft.  of  fluid  (Ibs/ou.ft) 
C  =  contraction  constant  of  orifice 

"herC  „   DA» 

C  »  

2gC» 

K   2C 
Combining  (2)  with  (1),  we  have  W,  »  —  —(b-x)  (3) 

ph  »r 

• 

If  now  we  assume  -—  to  always  remain  a  constant  C1 
Ph 

a    2C 

and  placing  c  C1  -  Co,  we  have  Wx  »  — a(b-x)   (4) 

mr 

which  is  an  equation  of  remarkable  physical  significance 
We  find  the  orifice  variation  to  be  a  parabolic 
function  of  the  displacement  and  is  quite  independent 
of  the  initial  recoil  velocity.  Therefore  with  the 
same  weight  of  recoiling  parts,  the  recoil  displace- 
ment is  practically  the  same  for  all  values  of  the 
initial  recoil  velocity.  Since  the  initial  velocity 
depends  upon  the  ballistics  of  the  gun,  we  may  com- 
pletely change  the  ballistics  of  the  gun  and  yet  with 
grooved  orifices  the  length  of  recoil  remains 
practically  unchanged. 

In  the  following  discussion  the  ratio  -r-  was  as- 
sumed to  remain  constant;  the  change 
in  the  length  of  recoil  depends  therefore  on  the 
change  in  the  ratio  H 

pb' 

Let  us  examine  this  ratio  for  the  change  under  two 
conditions, 

(I)     As  the  gun  elevates  where  the  weight 

component  is  brought  into  effect. 


438 


(2)     For  different  ballistics  of  the 
gun,  where  tbe  initial  velocity  is 

changed. 
Now  for  case  (1),        a  V* 

K  *  0.45  ^— -   and  assuming  the 

same  length  of 

recoil,  K  is  a  constant  and  independent  of  the 
elevation. 

If  K 

—  is  to  remain  constant,  its  reciprocal 
p 
h  must  remain  constant  for  all  elevations. 

Since  K  *  Pn+Fy+Rt-irrsin  6  where  Ph  =  total  hydraulic 

pull  (Ibs) 

Fv  =   recuperator   reaction    (Ibs) 
Rt  =  total   friction    (Ibs) 
Hence 

Ph       K-Fv-Rt+lfrsin  0  Fv+Rt-Wr   sin   £1 

—  *  •  =  1  -      '  ' 


K          K  K 

Since  Fv  and  K  remains  a  constant  for  all  elevations, 
in  order  that 

K  ph 

or  its  reciprocal  —  remain  a  constant,  we  must 

pb  K 

have  Rt  -Wr  sin  #t=  Rt  -Wp  sin  0g 

To  consider  extreme  conditions,  let  us  consider, 
horisontal  and  max.  elevation,  then 


where  0B  =  the  angle  of  elevation  at  max.  elevation. 
Now  Rt  »  R-  +  Rp  where  R*  =  the  total  guide  friction 
Rp  *  the  total  packing  friction 

Now  Rg  is  proportional  to  the  total  braking  *  K+lfrsin  0 
due  to  the  pinching  action  of  the  guides,  and  the 
packing  friction  remains  practically  constant  since  Pj, 

does  not  change  greatly.  Hence  on  elevation, 

Rt0m  >  Rt0o  usually  except  for  large  guns  with  balanced 

palls. 

From  actual  numerical  calculations  on  a  series  of  guns, 
tbe  term  Rto  was  found  to  be  slightly  greater  than 


439 


St0  ~*r  sin  ^M*  Therefore,  —  remains  practically 

ph 
constant. 

(1)     The  length  of  recoil  with  the  sane 


grooved  orifices  is  practically  in- 
dependent of  the  elevation  of  the  gun. 

In  case  (2)  with  different  ballistics,  we  have 
roughly,  Kt=*  0.45  mrV* 

K  *  0.45  mrV* 

g 

and  as  before  the  reciprocal  of  the  ratio  —  ,  becomes, 

ph  ^fy  '*rsin  0  h 

=-  a   1  --  therefore   for  a  constant   ratio, 

we   should   have, 

pv  +  fit-Wrsin  0  Py+Rt-lfrsin  0 

which  obviously  is 


K  K 

i  *        impossible. 

But  Fv+Rt-Wrsin  Of  is  always  small  compared  with  K, 
hence  the  difference  of  the  above  terms  must  be  cor- 
respondingly smaller. 

Hence  though  the  ratio  —  changes  with  different 

D 

ballistics,  the  change       h 
is  very  small. 

(2)     The  length  of  recoil  with  the  same 

grooved  orifices  is  practically  independent 
of  the  ballistics  of  the  gun. 


ROTI:    Tha  above  disoussion  on  length  of  racoil  ia 
retained  as  a  point  for  discussion.   The 
author's  conclusion*  are  not  however  well 
confirmed  by  observed  data.    Bditor. 


440 


COUNTER  RECOIL:     In  the  design  of  a  counter  recoil 
ELEMENTARY       system,  i»e  are  concerned  with  either 
DISCUSSION.      counter  recoil  stability  when  the 
gun  enters  the  battery  position  or 
with  the  buffer  pressure  in  the 

counter  recoil  regulator.   In  the  former,  we  are  con- 
cerned with  the  overall  force,  that  is  the  total 
force  towards  the  end  of  counter  recoil,  while  in  the 
latter,  with  the  c'recoil  buffer  or  regulator  re- 
action. Let 

Kv  -  total  resistance  to  counter  recoil   (Ibs) 
P7  »  total  recuperator  reaction         (Ibs) 
B^  =  counter  recoil  regulator  or  buffer  force  (Ibs) 
Rt  =  total  friction  (Ibs) 

wx  =  throttling  area  of  c'recoil  regulator  (sq.  in) 
C1  =  throttling  constant 
Afc  =  area  of  buffer  (sq.in) 
v  =  velocity  of  c'recoil  (ft/sec) 
The  critical  angle  of  elevation  for  counter  recoil 
functioning  is  at  horizontal  elevation.  Then  Kv=B£+Rt-Fv 
and  for  horizontal  c'recoil  stability  in  a  field  car- 
riage, we  have      w  ,   +  w  (b-x) 
K 


v 

h 
where  lg  =  distance  from  total  weight  of  system  to 

forward  overturning  point,  usua  lly  the  front 
\_^L  """""    wheel  base  (ft) 

x  *  displacement  in  c'recoil  from  out  of  battery 

position  (ft) 
b  *  length  of  recoil  (ft) 
h  =  height  of  center  of  gravity  of  recoiling  parts 

from  ground  (ft) 

We  may  express  Wslg   in  terms  of  the  static  load  on 
the  spade  then,  T  1  =  HglJ 

where  1  =  distance  between  spade  and  wheel  contact  with 
ground.  Then      T  1  +  Wr(b-x) 


where  T  =  150  to  200  (Ibs) 


441 


If  the  ground  is  assumed  to  exert  a  downward 
pressure  on  the  spade  comparable  with  the  load  T, 


Ky  =  0.85 


2T  1  +Wr(b-x) 


h 

which  gives  the  limitation  of  the  magnitude  of  the 
total  unbalanced  force  towards  the  end  of  counter 
recoil* 

For  simplicity  in  the  following  discussion  a 
constant  regulator  reaction  will  be  assumed  acting 
throughout  the  counter  recoi  I.  This  method  of  con- 
trol was  used  by  the  Rrupp  and  the  earlier  material  of 
the  Schneider  in  France. 

SPRIHG  RETOHH. 

Let  S  =  initial  or  battery  load  on  spring  column  (Ibs) 
Sf  =  final  or  out  of  battery  load  on  spring  column 

(Ibs) 
Ct  =  spring  constant 

Tbenp     .8      IT     -9 

Fvi  a  so>   Fvf  '  sf 

and  the  recuperator  reaction,  in  terms  of  the  c 'recoil 
displacement  x,  becomes, 

F-  »  S0  *  (b-x)=S0+S(b-x)  where  S  » =  the 

b  b 

spring  constant, 
dv 

then  mr  v  --  =  -K.. 
dx 

=  -(B£  +  Bt-Py) 
therefore 

Brv      i  Sx 

—  =  -  Bxx  -  Rt  x  +  (S0+S  b  -  — )  x 

2  * 

which  is  the  general  equation  of  c 'recoil,  with  a  con- 
stant regulator  reaction  and  spring  return.  When 
x  *  b,  v  =  0,  hence 

~Bxb~Rtb  +  (so+Sb"~  2""^b=0 
hence        _. 

B'  =  Sft+  ;:—  -  Rt(lbs) 
0  2 

This  same  value  may  be  obtained  by  a  consideration  of 


442 


the  potential  energy  stored  in  the  recuperator. 
The  potential  energy  of  the  recuperator,  becomes 

b         b   Sf-S 

*o  s  f  S0  dx  +  /   x  dx 

o         o     b 


sfso  b* 

V  *  —  T- 


b  (ft.lbs) 

2 

We   have,    then,    from   the  principle  of  energy, 

,        VSf 
W0  »  R^b+Bxb  =  «— b     since  Sf  =  S  b   +  So 

ou 

hence  B,     =  S_   +  —  -  R* 
x  o        g  L 

Substituting  this  value  in  the  energy  equation       * 

2  r 

and  siaplifying,  we  have  nrv  =  Sx(b-x)  hence  S  =   ' 

i          mrv*  b  (b-x)x 

..d  Bx  -(80-Rt)*  J^^- 


which  gives  the  value  of  the  constant  regular  reaction. 

Bx  *  -  Clbs)  where  C  =  the  reciprocal 

of  the  contraction 
factor  of  the  regulator  orifice. 

Ab  *  effective  area  of  buffer 

wx  =  variable  regulator  orifice,  and  since, 

»   S(b-x)x 
v  a  

»r 

C*Aw  s(b-x)x 


Bx 


175.rBx 


and   therefore   w2    =  _____   (fcx-x*  ) 

•»; 

Value  of 

of  regulator  (sq.in) 


443 

C  Ag  /-s 
where  CQ  =         * 

13.2  /  mrB' 
r  x 

BX  -  V  ^  -  Rt 

The  unbalanced  force  of  c'recoil,  becomes, 

dv     ,_« 
mr  v  -      •  (Bx  +  Kt  -  Fv; 

dx 

=  -  (S0  +  —  -  S0  -  Sb  +  Sx) 
2 

=  —  -  Sx  =  S(  -  -  x) 

2  2 

Hence  the  unbalanced  force  decreases  with  the  dis- 
placement of  c'recoil,  reverses  to  a  negative  value 
at  mid  stroke. 

The  initial  unbalanced  force  at  the  beginning 

of  c'recoil,  equals 

Sb   ,sf"so      sf"so 

*2  =  (~!b~      ~ 
The  overturning  force  at  the  end  of  c'recoil,  becomes 

Sb  _  sf~so 
.2      2 

GENERAL  EQUATIONS     The  functioning  of  counter  recoil 
OP  COUNTER  RECOIL,  may  best  be  studied  by  a  consideration 
of  the  physical  aspects  of  the 
dynamic  equation  for  counter  re- 
coil. Let 

pa  =  intensity  of  pressure  of  the  oil  in  the  air 

cylinder  (Ibs/sq.in) 
"ax  a  counter  recoil  throttling  area  between  air 

and  recuperator  cylinders  (sq.in) 
Ay  =  effective  area  of  recuperator  piston  (sq.in) 
KV  =  total  resistance  to  counter  recoil  (Ibs) 
Fv  =  actual  or  equivalent  recuperator  reaction 
at  any  displacement  "x"  from  the  out  of 

battery  position  (Ibs) 
wx  =  variable  buffer  orifice  at  c'recoil  dis- 


444 


placement  x  for  buffer  counter  recoil 

throttling  (sq.in) 

Then  during  the  counter  recoil  for  a  spring,  pneumatic 
or  similar  recuperator  system,  we  have, 

(1)  the  recuperator  reaction  acting  to 
displace  the  gun  forward  into  battery 
Fv  (Ibs) 

(2)  the  weight  component  resisting  Fy  -  - 
Wrsin  0   (Ibs) 

(3)  tlie  guide  friction  Rg  =  n  Wr  cos  0 
approx.  since  the  pinching  action  of  the 
guides  is  small  on  counter  recoil  and 

we  therefore  have  an  approxination  of 
pure  sliding  friction  throughout  the  greater 
part  of  counter  recoil.   This  reaction 
also  resists  Fv.  , 

(4)  the  packing  friction  Ks+p  resisting 
Fv  (Ibs) 

(5)  fhe  throttling  through  fhe  return  of 
the  recoil  apertures  together  with  the 
counter  recoil  buffer  throttling.  The 
throttling  through  the  recoil  is  small 
as  compared  with  the  buffer  throttling 
and  may  be  neglected  or  else  included 
with  the  buffer  throttling.  The 
throttling  is  proportional  to  the 
square  of  the  velocity  of  counter  re- 
coil and  inversely  as  the  square  of  the 
throttling  orifice,  that  is,  the  buffer 
braking  becomes, 

I   » 
I     COV 

H.  =  -     (Ibs)  and  resists  Fw 
*     a  «^________ 

% 
A 

Therefore,  w«  have 


n 

Fv-Wr(sin  0  *  n  cos  0)-R8+p  "  "" 


which  is  the  differential  equation  of  counter  recoil. 


445 

With  a  hydro  pneumatic  recuperator  system  it  is 
possible  to  regulate  counter  recoil  by  lowering  the 
pressure  in  the  recuperator  cylinder  for  the  greater 
part  or  the  entire  recoil,  by  throttling  the  oil 
through  an  orifice  between  the  air  and  recuperator 
cylinders.   Introducing  a  buffer  chamber  in  the  air 
cylinder  with  a  buffer  attached  to  a  floating  piston, 
gives  a  simple  means  for  varying  the  orifice  and  thus 
reducing  the  pressure  in  the  recuperator  cylinder  or  in 
the  recoil  cylinder  to  a  value  consistent  for  the 
proper  movement  of  the  recoiling  parts  in  counter  re- 
coil . 

The  pressure  in  the  recuperator  cylinder  due  to 
throttling  through  the  orifice  between  the  air  and 

recuperator  cylinders,  becomes, 

iii  a 
,   co  v 
Pv  s  Pa  -   W2 
wax 

Hence,  for  the  motion  of  ths  recoiling  parts  in 
counter  recoil,  we  have, 

'  2 

PvAy  -  Wr(sin  0  +  n  cos  0)  -  Rs+p  -   "g   =  mp  v  -— • 

wx  dx 

or  substituting    for  pv,    we   have 

Cn  i 

.  C       2  dv 

Pa-Ay  -  Wr(sin  J0+n  cos   &)-  Rs+p~    (— •*•  *  ~~ ^)v  =   airv  — (2 

w?.,        w?  dx 


where  Co  =  A^o'  ' 

i 

Now  p   AV  may  be  regarded  as  the  equivalent  recuperator 
reaction,  that  is    Fv  =  pa  Ay   and  further  assuming 
the  regulation  to  be  entirely  effected  through  the 

throttling  in  the  recuperator,  we  have,  for  eq.(2) 

n 

Crt  A  ,r 

Fv  -Wr(sin0+n  cos  0)-Rs+_  -  —  v?=mrv  —   (3) 

"ax       dx 

which  is  exactly  similar  to  the  previous  equation  of 
counter  recoil  for  a  simple  spring  recuperator  system. 
The  external  force  on  the  total  mount,  is 


446 

i  dv 

Kv  »  »r  v  —  ,  together  with  the  weight  of  the 

recoiling  parts  Wp. 
During  the  acceleration, 

Kv  =  mr  v  -—  acts  towards 


the  breech,    and 
during    the   subsequent   retardation, 

KV  *  Br  v  d7  aots 

towards 

the  nuzzle.  During  the  acceleration  Ky  is  necessarily 
always  less  than  K  the  total  resistance  to  recoil  since, 

~Cv* 
K  =  F..+R  +  -T—  -  W_  sin  0,     for  the  recoil  and 

w*  —  —  —  —  —  _—  . 

cV 

Ky  =  Fv  -R  -  —  j—  -  Wr  sin  0,  for  the  counter  recoil, 
wx  '  ' 

therefore  2    i  2 

Cv    C  v 
K-KV  =  2R  +  -j-  +  —  —  ,  roughly  assuming  total 

friction  the  sane  on 

recoil  and  counter  recoil.  Hence,  in  the  design  of  a 
counter  recoil  system  we  are  only  concerned  with 
counter  recoil  stability,  and  not  at  all  with  the  re- 

action during  the  acceleration.   If  we  let,  further, 
Ws  =  weight  of  total  system  (Ibs) 

ls  -  horizontal  distance  fron  front  hinge  or  con- 

tact of  wheel  and  ground  to  the  center  of 
gravity  of  the  total  system  in  battery  (ft) 
C   =  constant  of  counter  recoil  stability 

Overturning  counter  recoil  moment. 

Stabilizing  counter  recoil  moment. 
i 
d  =  perpendicular  distance  from  front  hinge  or 

contact  of  wheel  and  ground  to  line  of 
action  of  Ky  through  center  of  gravity  of 
recoiling  parts  (ft) 

then,  for  stability  at  angle  of  elevation  6,  we  have 

0 


s  +  Dr  cos0)-Fv    = 


(2) 


447 

dv      w-3l3+Wr(b-x)cos  0 

and  -  •-  v  — •  =  C  t 3         (3) 

dx  d1 

which  gives  us  the  velocity  curve  against  displace- 
ment consistent  with  counter  recoil  stability.  Sub- 
stituting v  in  (2)  enables  us  to  determine  the  variable 
orifice  wx  consistent  with  counter  recoil  stability, 
since  Fy  is  a  known  function  of  x. 

During  the  acceleration,  we  have 

°;>*   t, 

Pv-Wr(sin  0  +  n  cos  0)  -  Rs+p  ~  "    s  Br  v  — 

and  since  we  are  not  concerned  with  stability,  for 

•  inisuiB  time  during  the  acceleration  Ky  should  be  made 

a  maximum,  that  is  the  hydraulic  braking  tern 
should  be  made  zero,  hence 


cV 


dv 
Fv-Wr(sin  0+  n  cos  0)  -Rs+p  =  »r  v  —- 

Let  further  vm  =  aaximun  velocity  of  counter  recoil 

(ft/sec) 
xm  =  corresponding  displacement  to 

maximum  velocity  from  out  of  battery 

position  (ft) 

Then,  for  ideal  counter  recoil,  that  is  the  counter 
recoil  functioning  in  nininun  time  and  consistent  with 
stability,  we  have, 

o  i   b 

~  /   «r  v  dv  =  —  /  [Wslg*Wr(b-x)  cos  0)dx     (5) 


from  which  we  obtain, 
a  • 

Brvm    C 

«      *  « 
To  determine  xn,  we  have 


448 


/•x»  vm 

'Fv   "  wr(sin  Of  +    n  cos   0)-  Rs+p]dx   =   /          nir   v   dv 


hence 

X 


.  IE  P 

Fv  dx  -  [Wr(sin0  +  n  cos  0)+H8+pl« 


—  —  -  cos  0  ]  (6) 

2 

and  knowing  Fv  as  a  function  of  x,  we  may  solve  for 
xm.  Substituting  in  (5')  we  easily  obtain  vm  which 
gives  the  maximum  velocity  of  counter  recoil. 

Thus  we  see  during  the  acceleration  it  is  de- 
sirable to  make,  Kv  a  maximum,  that  is 

Kv    =  Fv-Wr(sin  0  +  n  cos  0)-  Ro+n 
vmax 

and  during  the  retardation  Ky  should  be  consistent 
with  counter  recoil  stability,  that  is 


dv 


cos 


which  can  be  obtained  by  increasing  the  buffer  or 
counter  recoil  regulator,  such  that, 

G'V*  W  gl'+W  (b-x)  cos  0 

0+W  cos  0)-F  =  C[  -  -  -  ] 


A  simple  graphical  solution  of  the  above  analysis  may 
be  made  as  follows: 

Lay  off  the  recuperator  reaction  Fvf-Fv^  and 
from  the  ordinates  of  this  curve  subtract  Wr(si"n  J0  •»• 
W  cos  0)+Rs+_  which  gives  the  unbalanced  reaction 
proportional  to  the  ordinates  to  AB,  during  the  ac- 
celeration period.  Draw  in  below  00',  CD  parallel  to 
the  counter  recoil  stability  slope  Q  R,  such  that 

—  =  —  =  C  ,  the  constant  of  counter  recoil  stability 

assumed.  Then  we  locate  M  such  that 
the  area  OABM  =  area  M  Of  D  C.   Since  OABM  is  pro- 


449 


w/m 


+VJC0S  t)  y-  >PJ  +  p 


C  'XECO/L  ENEffGY  PL  OT5 

COS 


MO  'DC 


450 

oortional  to  the  work  done  during  the  acceleration, 

we  have 

Area  0  A  P  M  =  -  Mp  Vm 

• 
The  velocity  curve  may  be  constructed  graphically 

since  any  increment  area  abed  is  proportional  to  the 
change  of  kinetic  energy,  that  is 

a.      i*  a 
Area  a  b  c  d  I  -  mr(vt-vj) 

and  thus  knowing  the  previous  velocity,  we  may  con- 
struct a  velocity  curve  directly. 

The  energy  equation  of  counter  recoil: 

The  dynamic  equation  of  counter  recoil,  is 

cV 

Fv-(n  cos  18  +  sin  0)Nr-Rs+p —  *  mr  v  -— 

wx        dx 

where  Fv  =  the  recuperator  reaction 
Rg+   =  total  packing  friction. 

& 

=  hydraulic  buffer  resistance 

"* 

x  CQv  v 

Integrating,    we    have  /    (Fv-»rsin0-Rt-  — y— )d**/   mrv  dv 

o  wx  o 

where  Rt    =  a  W_  cos  0  +  ^s  +  n  ,    , 

x  x    C0v  y2 

Separating,   we   have  /  Fydx-(Wrsin0+Rt )X-/     — - —  dx=mr  r- 
o  o     wx  2 

FJow   since   the   relative   energy   in   the   recuperator,    de- 
pends  only   on    the  position   in   the    recoil,    we    have, 

dW 
Fw  »  -  —       since  ¥v  dx  =  -  dW 

de 

where  W  is  the  relative  potential  energy  of  the  re- 
cuperator, which  is  equal  to  the  work  of  compression 
(approximately)  for  a  displacement  in  the  recoil  (b-x)(Fy) 

If  W  =  the  potential  energy  of  the  recuperator  in  the 
out  of  battery  position, 


451 


•                                                                                i    i  a 

r   *    dW                                                           Cov  mrv 

-     /       —  .   dx  -  (ff-sin  0+Rt)x  -  -2—  dx  =  -£— 

W        dx                                                         w  2 


from  which  we  obtain 


_ 

CQV 


(*t-  Wx)-(«rrsin  2T+Et)x  -  /  -  dx  )=  - 

"x        2 

which  is  the  general  energy  equation  of  counter  recoil 

Obviously  at  any  displacement  in  the  counter  recoil  x, 

-  '   «  * 

CQv  rarv 

If,    +(Wrsin  0+Rt)x+  /  -  dx    +  -  =   W.      a  constant 

**  **  4  f\  O        MH^HWMMM^^HMM^^^K 

»x       2 

That  is,  the  total  energy  at  any  recoil  x,  is  divided 
into  the  potential  energy  of  the  recuperator,  the  work 
done  by  friction,  the  work  done  by  buffer  throttling 
and  in  the  kinetic  energy  of  the  recoiling  mass. 

Between  any  two  displacements  in  the  counter  re- 
coil K^  and  xa  we  have,  approximately,  provided  the 
points  are  sufficiently  close: 


which  gives  us  a  method  of  computing  vx   knowing  vx 
from  the  previous  point. 

COMPUTATION  OF         With  a  given  set  of  counter 
COUNTEE  RECOIL.     recoil  orifices,  the  velocity 
and  force  curve  of  counter  re- 
coil may  be  calculated  by  either  of 
the  two  following  methods: 

If  Fv  =  actual  or  equivalent  recuperator  reaction  at 
any  dis  placement  "x"  from  the  out  of  battery 
position  (Ibs) 

F?i  =  initial  recuperator  reaction  (Ibs) 
wx  =  variable  orifice  for  counter  recoil  throttling 
at  displacement  "x"  from  the  out  of  battery 
position  (sq.in) 
CQ  =  counter  recoil  throttling  constant 


452 


n  =  coefficient  of  guide  friction 
Rs+p  =  total  c'recoil  packing  friction  (Ibs) 
Ay  =  effective  area  of  recuperator  piston  (sq.in) 
VQ  =  initial  volume  of  recuperator  (cu.in) 
x  *  counter  recoil  displacement  (ft) 

METHOD  I  -    LOGARITHMIC    METHOD. 


The  dynamic  equation  of  c'recoil,  becomes 

cV 

Fv  -  Wr(sin  0  *  n  cos  0)-Rs+ —  =  mr  v  — 

"x         d* 

If  we  let,  R  =  ffr(sin  0+  n  cos  0)+Hs+p 

GovZ       dv 

then  F  -  R =  m_v  — - 

wx       dx 

Now  Fy  and  wx  are  both  functions  of  x  and  therefore 
the  equation  cannot  be  readily  integrated.   If, 
however,  we  take  a  small  interval  Fv  and  wx  may  both  be 
assumed  constant  during  this  interval.   Considering 
any  two  points  x^  and  xa  in  the  counter  recoil, 
we  have 


*a       v2   nrv  dv 

dx  =  /    where  A  =  Ftf-R 


Rearranging,  we  have 

C0v» 

d(A-  -2— 
x  *        w*  i 


2°i     o!,' 

A— 


hence  integrating,  we  find 


453 


and 


*  loge(A  - 


m 


r  "x. 


therefore 


.  .      o2t 

log  (A  --  )'  log  (A  --  )  -- 

wx'  "x1      2.3nrw«i 

2  a  *s 

where  A  =  Fv-Wr(sin  0+  n  cos  2J)-Rs+p    (Ibs)  fro« 
which  ire  nay  construct  the  velocity  curve. 

The  advantage  of  this  method  is  that  a  small 
variation  of  Fv  and  v»x  has  a  negligible  effect  on  the 
equation  of  motion  and  therefore  fairly  intervals  nay 
betaken  provided  the  throttling  orifice  of  counter  re- 
coil is  not  changing  rapidly.   During  the  buffer  period 
where  the  throttling  changes  rapidly  small  intervals 

oust  be  taken. 

The  total  unbalanced  force  acting  on  the  recoiling 

parts  during  counter  recoil,  is 

dv         Av          , 
mr  v  —  =  mr  v  —          (approx.) 

From  this  the  unbalanced  force  (total  accelerat- 
ing or  retarding  force) 

Fv-Wr(sin  0  +  n  cos  (?  )-Rg+p 


x 
•ay  be  calculated  and  plotted. 

To  compute  the  recuperator  reaction  at  any 
point,  we  have  for  spring  recuperators, 

srso 

Fv  =  S0  +  -^-£-(b-x) 
b 


454 


where  SQ  *  initial  or  battery  spring  reaction  (Ibs) 
Sf  =  final  or  out  of  battery  spring  reaction 

(Ibs) 
and  for,  pneumatic  or  hydro  pneumatic, 

Vo        k 
FV  '  Pai  V  V0-12AV  (b-x) 

V        k 
V0 


where  b=  length  of  recoil  (ft) 

x  =  c 'recoil  displacement  from  out  of  battery 
position  (ft) 

VQ=  initial  volume  (cu.in) 

To  compute  Rs+p,  we  have,  RS+D=100  to  1-50  Zd  for 
ordinary  packing 

where  d  =  iiam.  of  any  one  of  ths  various  recoil  rods 
(in) 
Rs+p=Z(Ct+Cap)»Z[0.15(.05  *WpdpPlBa3t)+0.75(.05TiWpdpp)] 

(Ibs) 
where  w  =  width  of  the  various  packings  (in) 

dp  =  diao.  of  the  annular  contacts  of  the  various 
packings  (in) 

pmax  =  the  design  pressure,  usually  the  max. 
pressure  in  the  cylinder  to  which  the 
packing  is  subjected  to   (Ibs/sq.in) 

p  =  actual  pressure  during  the  various  points 
in  the  counter  recoil  to  which  certain 
parts  of  the  packing  are  subjected  to 
(Ibs/sq.in) 

Obviously  since  p  is  variable,  Rs+p  must  be  variable 
daring  the  counter  recoil  but  aq  average  value  of  p 
•ay  be  assumed  and  the  corresponding  Hg4p  can  be  used 
with  sufficient  accuracy.  t 

I V 

Computation  of  the  throttling  resistance  C0— 


455 


(1)  with  a  filling  in  buffer,  the 

counter  recoil  regulation  being  effective 
throughout  the  counter  recoil: 

we  may  neglect  the  small  throttling  through  the 

apertures  of  the  recoil  orifice,  and  then, 

*     Cf2A!v* 
i  v        Abv 

co  T  (Ibs) 

175w» 

where  C'  =  the  reciprocal  of  the  throttling  constant 
Ab  *  area  of  the  buffer  (sq.in) 
wx  -  buffer  throttling  area  (sq.in) 

(2)  with  some  form  of  spear  buffer, 

the  buffer  action  being  effective  only 
during  the  latter  part  of  counter  re- 
coil, 
we  have  three  stages: 

(a)  the  void  displacement  with 
no  regulation. 

(b)  throttling  through  the  recoil 
apertures  which  cannot  be 
neglected  due  to  the  much  higher 
velocity  of  c 'recoil  than  in 
case  (1). 

(c)  throttling  through  the  buffer 
orifice,  the  throttling  resistance 
being  large  as  compared  with  the 
resistance  due  to  throttling  through 
the  recoil  orifice,  the  latter 
being  neglected. 

In  (b),  we  have, 

*  f  .      »3   * 

(A+ar)  v 


co  _7 


175  w$ 


xr 


where  A  =  effective  arc  of  recoil  piston  (sq.in) 

ar  =  area  of  recoil  rod  (sq.in) 

wxr~  area  of  recoil  throttling  grooves  (sq.in) 


In  (c),  we  have,  as  in  (1) 


456 


•    «*  »'  * 

,  /  B  C  V 

°  w«    175  w» 

where  Ab  =  area  of  buffer  (sq.in) 

wx  =  buffer  throttling  area  (sq.in) 
With  a  hydro  pneumatic  recoil  systen, 
In  this  type  it  is  possible  to  loner  the  pressure  in 
the  recuperator  by  throttling  through  a  constant 
orifice. 

Now  it  has  been  shown,  that 


At  the  end  of  recoil  if  a  spear  buffer  in  the  recoil 
brake  cylinder  also  functions, 


o 

y  =  the  effective  area  of  the  recuperator 
piston  (sq.in) 

w0  a  the  c1  recoil  throttling  area  between  the 
air  and  recuperator  cylinders  (sq.in) 

METHOD  II  -  THB  HHBRGY  KUTHOD. 

From  the  energy  equation,  we  have,  for  any 
arbitrary  interval, 


»    a 

Vx2~vx 
(Wx»-Wx2)-(Wrsin  0+Rp)(w2-xi;  --  -  (x,-xt  )»mr(  -  ; 


i  a 

Cov 


where  Wxn  -  the  recuperator  potential  energy  at  the 

point  "n"  in  the  counter  recoil  (ft.lbs) 
To  compute  Wxn  we  proceed  as  follows, 
With  a  spring  recuperator, 

• 

o   —  O 

"xn  =  -^[so  *  ^  —  —(b-x)J  d(b-x)  (ft.lbs) 
b 


457 


=  S0(b-x)+  5f  °  (b-x)'    (ft.lbs) 

2b 

where  So  =  initial  spring  recuperator  reaction  (Ibs) 
Sf  =  final  spring  recuperator  reaction  (Ibs) 
b  =  length  of  recoil  (ft) 
x  =  displacement  in  counter  recoil  (ft) 

With  a  pneumatic  or  hydro  pneumatic  recuperator, 


b-x  b-x          V0  k 

**n  =  *          Fvd(b-x)=F¥i  /          (  -  )     d(b-x)    (ft.lbs) 
o  o  V0-AY(b-x) 

where  k  =   l.Koil   in  contact  with  air) 

=  1.3  oil  and  air  separated  by  floating  piston 

or  pure  pneumatic) 

AO  =  effective  area  of  recuperator  (sq.ft) 
VQ  =  initial  volume  (cu.ft.) 
Pvi=  initial  recuperator  reaction  (Ibs) 
Integrating,  we  have 


xn 

Av(k-l)      y"-1          V*-1 
where   V   =   Vo-Av(b-x).      Further   since, 

Pai   Vj  =   PaVk       or  ^..    A" 
Pai        V 

then,  pai          ,         ,    k 

k    1 


P      V  -  PiV 


aio 


Hence,  when  V  is  in  cu.  ft.,  Av  in  sq.  ft.  and  b-x  in 
ft,  we  have 

V  =  V0-Av(b-x)     (cu.ft) 


Vo  k 
Fv  =  F>vi^         (lbs) 


Wx=  /    ;;        (ft.lbs) 
Av(k-l) 


458 


Usually  it  is  more  convenient  to  express  V  is  in  cu.  in., 
Ay  in  sq.  in  and  b-x  in  ft. 

V  =  V0-12A?(b-x)      (cu.in) 

*V  Fyi(— )  (Ibs) 

V 


12Av(k-l) 


(ft.lbs) 


To  compute  Fy,  we  have  log  =  k  log  — ,   a  linear 

Fvi        v    logarithmic 

equation  and  therefore  may  be  readily  plotted.  There- 
fore, vie  may  make  a  table  for  computation  of  the 
potential  energy  of  the  recuperator  as  follows: 


V                                        wx 

X 

12Av(b-x) 

V 

Vo 

F 
rv 

F  V-F    •  V 

r  VT     rVl  VO 

K     lOg    ^ 

12Av(k-l) 

459 


We   have,    from   the   energy  equation 
W0-Wxi-(Wrsin0+Rp)xt • 


+m_  — 


s    », 


O      2  2  3 

Hrt-W_s-W_sinjft+RD)(x  -x    )-  — —  +   mr  —  =  m.  — 

XX  r  3         *  _2 .  *0  O 


n-1 


'xn 


The   solution   of   these  equations,    may  be   put   in   a   table 
form: 


X 

», 

WX,-V»X2 

(Wrsin0+Rp)Ax 

"x 

^y 

2 

Vn-l 

X 

n 

v 

o 

»0 

«, 

Wxi 

«0-«». 

(Wrsin<?+Rp)xt 

«x' 

m^ 

v, 

«, 

^ 

l^!.. 

(Wrsini?+Rp) 
(x,-xt) 

X2 

C  v 

_  o    t 

2 

"x' 

2 

2 

V 

"•""I 

m  i 

', 

«, 

v 

V-v 

(xa-x2) 

wx2 

c'v; 

wSz 

X3 

• 

2 
V 

r2~ 

2 

m  ^ 

: 

xn 

o 

WX(n-l) 

(Wrsin0+Rp) 

"xn 

C0v2n-l 

"xn 

o 

o 

From   the   above   table   we   may   plot   the   velocity  curve, 
To  obtain   the   unbalanced   force    (accelerating   or   re- 


460 


tardation  force  of  c'recoil)we  have, 

cfv*      v*  _va 
Fv-Wr(sin  0+n  cos  0)~Rp =  mr(— -)   (Ibs) 

RELATIVE  ADVANTAGES  OF  THE  LOGARITHMIC  AND  ENERGY  METHOD 
FOR  COMPUTATION  OF  COUNTER  RECOIL: 

In  the  design  and  computation  of  a  c'recoil  system, 
we  are  either  concerned  with  counter  recoil  stability 
which  is  the  primary  limitation  on  c'recoil  for  small 
caliber  mobile  carriages,  or  with  the  maintaining  of 
a  low  and  constant  buffer  pressure,  where  c'recoil  is 
no  longer  a  consideration  and  the  potential  energy  of 
the  recuperator  is  large,  as  in  large  caliber  artillery. 

In  the  former  case,  it  is  import-ant  that  the 
total  unbalanced  resistance  to  c'recoil  or  the  total 
retardation  towards  the  end  of  counter  recoil,  either 
remain  constant  or  follow  the  c'recoil  stability  slope. 
In  the  latter  case,  however,  it  is  important  to  maintain 
as  low  buffer  pressure  as  possible  and  thus  a  constant 
buffer  resistance  is  used  in  spite  of  the  total  resist- 
ance to  c'recoil  rising  towards  the  end  of  c'recoil. 
In  the  calculation  of  the  total  accelerating  or  retard- 
ing force  in  c'recoil,  the  logarithmic  method  and  the 
simple  dynamic  equation  of  c'recoil  are  preferable  since 
we  are  only  concerned  with  the  total  unbalanced  force 
on  the  recoiling  mass.  During  the  first  period  of  c'recoil 
a  constant  throttling  orifice  is  usually  used  for  reg- 
ulation and  large  intervals  may  be  taken  by  the 
logarithmic  method.  During  the  retardation  the  total 
resistance  to  c'recoil  is  usually  constant  and  there- 
fore we  have  the  simple  dynamic  relation  of  a  mass 
being  brought  to  rest  by  a  constant  force.   With  a  con- 
stant buffer  force,  the  energy  method  is  preferable 
since  the  work  done  by  the  buffer  and  corresponding 
kinetic  energy  and  therefore  the  velocity  of  c'recoil 


461 


can  be  quickly  estimated. 

Estimation  of  the  buffer  resistance  of  c'recoil, 
with  constant  buffer  force  and  corresponding 
velocity  of  c'recoil: 

(1)     If  the  buffer  force  acts  only  during 
the  latter  part  of  c'recoil,  we  have, 
three  periods: 

(a)  the  accelerating  period,  cor- 
responding to  the  void  displace- 
ment. 

.....  •  .   ;  .   r-  .:*<c*/w  8OJ   TO: 

9 
*a 

(W0-Wa)-(Wrsin  0+Rp)Xa=  mr  — -    (ft.lbs) 

(b )  the  retardation  period  where 
throttling  takes  place  in  a  re- 
verse direction  through  the  re- 
coil apertures  only. 

I   2 

(W  -wb)  -  (WrsinlZf+RD)(Xb-Xa)-  /c!   -^-  dx  =  —  (vj  -  va) 

H   "  v        w  ^  9 

*a    "x 

xb  ^ov 
If  we  neglect  the  term,  /   — ~  dx  as  small,  we 

xa          have  immediately 

2 

(ffo~ffb)~(Hrsin  0+Rp)Xb=mr  —»   (ft.lbs) 

(c )  the  retardation  period  where 
the  running  forward  brake  or 
c'recoil  buffer  comes  into  action: 
assuming  a  constant  buffer  force, 
we  have  „  i  2 

~fr  =  Bx 


and 


2 

r  — 
2 


462 


* 

vb 
Substituting  for  (-mr  — •)  from  the  previous  equation, 

we  have 

Bx(b-xb)*W0'  Ofrsin  0r+Rp)b 

be  nee 

Wp-(W   sin  0*R_)b 

BX»   — E —  <lbs) 

b-*b 

If  Ab«  the  area  of  the  buffer  (sq.in) 
db»  b-xb  »  length  of  the  buffer  (ft) 

b  »  length  of  recoil   (ft) 
pb*  the  average  buffer  pressure  (Ibs/sq.in) 
then  we  have  for  the  average  buffer  pressure, 

W0-(Wrsin 

p  * 


where 

wo  *  ' pvf  *  "  Fvi 

Vf3Vo-Avb  m  »  ratio  of  compression 

To  compute  tbe  velocity  curve  during  tbe  buffer  action, 
we  have 

x  G'V*      *r  •  « 

tfb»Wx-(Wpsin  Or+Rp)(x-xb)-  /   -^-dx  -  ^~(vx-vb) 

xb   "x 

Since  Wx  and  v  vary  at  each  point,  tbe  above  equation 
•ay  be  divided  into  a  step  by  step  process,  i.  e. 

Wb-Wx,-(Wr«in  0+Rp)(xt-xb)-  j~-  (xt-xb)=  -|  (»Ji-vJ) 

^I(x  -x  )-  i  v',-v' 


463 


Pron  the  velocity  curve  and  buffer  pressure  pb 

(2)     Where  the  buffer  force  acts 

throughout  the  c'recoil. 

At  the  beginning  of  c'recoil,  the  recoil  apertures 
are  snail  and  the  throttling  through  them  during  the 
c'recoil  cannot  be  neglected.   Since,  however,  this 
additional  throttling  is  effective  only  for  a  short 
distance  at  the  very  beginning  of  counter  recoil  we 
have  as  a  close  approximation,  for  the  average 
buffer  force  of  c'recoil  (assumed  constant) 

WQ-(WTsin 


where  as  before,  WQ=  A  (k,1) :   %f  •%!« 

COUNTER  RECOIL  SYSTEMS.     Counter  recoil  systems  may 

be  broadly  classified  into: 
(1)  Those  in  which  the  brake 
comes  into  action  during  the 
latter  part  of  counter  recoil. 

(2)  Those  in  which  the  brake  is  effective  throughout 
the  counter  recoil. 

With  (1)  we  have,  usually  some  form  of  spear 
buffer  which  comes  into  action  towards  the  end  of  re- 
coil. 

With  (2)  we  have,  usually  a  "filling  in"  type  of 
buffer,  the  buffer  being  filled  during  the  recoil  and 
acting  throughout  the  counter  recoil. 

Type  (2)  gives  obviously  far  better  counter  re- 
coil regulation  than  with  type  (1)  where  in  the  latter, 
we  have  considerable  free  counter  recoil  and  corres- 
ponding high  velocities  before  the  buffer  action  takes 
place.  This  is  especially  true  for  long  recoil  guns. 

(1)     Counter  recoil  systems,  where  the  brake 

is  only  effective  during  the  latter  part 
of  counter  recoil.  The  counter  recoil 


464 


functioning  may  be  divided  into  three 
periods: 

(a)  The  acceleration  period  during 
the  void  displacement. 

(b)  The  retardation  period  where 
throttling  takes  place  in  a  reverse 
direction  through  the  recoil 
apertures  only. 

(c)  The  retardation  period  where 
the  running  forward  brake  comes 
into  action. 

During  period  (a),  we  have 

Fv-Rs+p-Wr  (sin  0  +  n  cos  0)  =  rar  v  —  -  and  the  void 
displacement      .  v 

x  =  !£   (ft) 

03     A 

where  aP  =  area  of  recoil  rod  (sq.in) 

A  =  effective  area  of  recoil  piston  (sq.in) 
b  =  length  of  recoil  (ft) 

hence 

f  oa  mrva 

--  -  - 


0      [Fv-Rs+p-Wr(sin0  +  n  cos 
As  an  approximation,  we  have 

Fvf+Fva  *a  xa   mrva 

(  -  )    -  [Rs+_+Wr(sin  0  +  n  cos  0)]    =  - 

2  2 

where   Fvf   =    the   max.    recuperator   reaction 

pva   =    the    recuperator   reaction  at    the  end   of 
the   void  displacement. 

(Fvf+Fva)-2(Rg+    +Wr(sin0+n   cos0)]Xa 

f  - 
mr 

(ft/sec) 

where  va  is  usually  the  max.  velocity  of  counter 
recoil.  During  period  (b),  we  have 

t 

Fv  ~  Rs+p~ffr(sin  9  +  n  cos  0)  --  -  —  =  m  v  — 

w^        dx 


A 
* 


465 


V 

where  — —      =    hydraulic   braking    reaction  due   to 
throttling   through   the   recoil 

iiii  '  "fii  *»    r,n  '  '  - 

apertures. 

Now  the  constant  CQ  is  different  from  that  of 
recoil  since  the  area  of  displaced  fluid  and  con- 
traction of  orifice  on  the  return  motion  are  dif- 
ferent from  these  factors  in  the  recoil.   However, 
for  a  first  approximation,  we  may  assume  CQ  the  same  both 
in  recoil  and  counter  recoil.   If  Va  is  the  velocity 
of  recoil,  with  total  hydraulic  pull  P^  at  displace- 
ment b  -  XQa  in  the  recoil,  we  have 

2       2 

C0v    va 

s  "^7  Pb  approximately 

and  therefore,  approximately, 

Fva+Fvb  va 

(— )  -  Ks+p-Hr(sin  0+  n  cos  0)  -  —  Ph(xb-xa)  = 


where  Fva  =  recuperator  reaction  at  end  of  the  void 
Fyjj  =  recuperator  reaction  at  entrance  to 

buffer. 

Prom  the  above  equation,  knowing  va  we  may  readily 
compute  V{j»  During  period  (c),  we  have 

2      _  I   2 

CQv    CQv         dv 

-V?r(sin0+n  cos  0)   -  — - —  -  — —  3   mr  v  — 
wh  "x  dx 

where   — —  =   the    hydraulic  braking  due    to   the 

W 

counter  recoil  buffer. 

Now  the  term  n  *                          n '  2 
o                          °o 
'  •     is  small  compared  with  

"6  u' .  *x 

and  may  be  neglected,  especially  during  the  latter 
part  of  period  (c). 

Period  (c)  is  the  critical  period  of  the 
counter  recoil  since  the  reaction, 


466 


-fl —   *   Wr(sin  0+n  cos  0)+R3+p-Fv   =  — — 
t 

where  =   the   max.    stabilizing    force   in  battery. 

d1 

The  buffer   throttling   should   be   designed,   either  with 

cov*  ,   "sis 

-Kv  =  — r~   *   Wr(sin  0  +    n   cos  0VRg+  -Fv=   Cs  — - 

d1 
a  constant   or 

r '»*                                                              w   1 ' 
uo                                                                i     SAS 
-   Ku   =  — —  +  H_(sin0«-n  cos0)+R.4.n-F_=C(,  ( —  )+Wr  (b-x)cos  0 

V  *  *  2»Tp         »          o  I 

"x 

that  is  consistent  with  the  stability  slope  of  counter 
recoil.   By  the  latter  method  the  buffer  action  nay 
have  a  somewhat  shorter  displacement  in  the  recoil  and 
yet  maintain  the  same  factor  of  stability  as  in  the 
former. 

At  this  point  it  is  well  to  emphasize,  that  a 
constant  buffer  resistance  is  entirely  inconsistent 

with  counter  recoil  stability  since  the  total  counter 

recoil  resistance  becomes  greater  in  the  battery 
position  than  at  the  entrance  to  the  buffer.  Therefore, 

a  longer  buffer  is  required,  for  the  same  mean  re- 
sistance to  counter  recoil. 

(2)     Counter  recoil  systems,  where  the 

brake  is  effective  throughout  the  recoil. 

In  this  type  of  counter  recoil, 

it  is  customary  to  regulate  the  maximum  velocity  at- 
tained during  the  acceleration  period  to  a  low  value, 
by  the  use  of  a  constant  orifice  throughout  the  ac- 
celerating period.   A  constant  orifice  during  the  first 
period  of  c 'recoil  has  distinctive  advantage  since  it 
gives  a  satisfactory  control  together  with  simplicity 
from  a  fabrication  point  of  view. 

During  the  latter  part  of  the  counter  recoil,  it 
is  obviously  necessary  to  introduce  a  variable  orifice 
in  order  that  the  recoiling  mass  may  be  brought  to  rest 


467 


gradually.   We  have,  therefore, 

(a)  The  accelerating  period  with 
a  constant  orifice. 

(b )  The  retardation  period  with  a 
variable  orifice. 

We  have  the  two  systems  of  regulation: 

(1)  By  a  buffer  brake  control  in  the 
recoil  hydraulic  brake  cylinder 
throughout  the  counter  recoil. 

(2)  By  lowering  the  pressure  in  the  re- 
cuperator cylinder  by  throttling  through 
an  orifice  between  the  air  and  re— 
cuperator  cylinders. 

With  simple  spring  or  pneumatic  recuperator  systems 
we  must  use  a  regulation  system  similar  to  type  (1). 
With  hydro-pneumatic  recoil  systems  we  may  use  type  (2) 
alone,  as  in  the  St.Chamond  or  Puteaux  brakes,  or  a 
combination  of  type  (1)  and  type  (2)  regulation,  as 
in  the  Filloux  and  Vickers  recoil  mechanisms. 

In  either  type  (1)  or  type  (2)  regulation,  for 
the  running  forward  brake  effective  throughout  counter 
recoil,  we  have,  exactly  the  same  characteristic 
dynamic  equation. 

With  a  simple  recuperator  of  a  s-pring  or  pure 
pneumatic  type,  we  have  for  the  equation  of  motion, 

CV     dv 

Fv~Wr(sin  0  +  u  cos  0)  -  Rs+p —  =  mr  v  -— 

"x         ax 

whereas  with  a  hydropneumatic,  assuming  the  pressure 

lowered  in  the  recuperator  cylinder  by  throttling 
between  the  air  and  recuperator  cylinders,  we  have, 

_  I  I  I   2 

i   co  v  dv 

<Pa  ~  ~ )Av-Wr(sin  0+u  cos  Of)  -Rs+p  =  mr  v  — 

and  if  we  let  pa  AV=FV«  the  equivalent  recuperator 
reaction 


468 


_  i  i  i  *          - "  a 

C0  *         cov 

— g  "•  -  •  •  Ay  =  — —   we  have,  as  before 

wtr  wv 

-."    « 

cov  dv 

Fv-Wr(sin  0+u  cos   0)-Rs+p  -         ••    =   mr   v  — 

w*         dx 

Further,  for  the  critical  condition  of  counter  re- 
coil stability,  that  is  counter  recoil  at  horizontal 
elevation,  in  type  (1),  the  reaction  on  the  carriage, 
consists  of: 

(1)  Fy  acting  to  the  rear 

(2)  in  Wr  +  Ro  +  n  acting  forward 


— ~  acting  forward 

It  may  be  easily  shown  that  the  resultant  of 
these  reactions  aots  in  a  line,  through  the  center 
of  gravity  of  the  recoiling  parts,  the  effect  of  the 
reaction  of  the  guides  being  to  transfer  the  various 
resistances  and  pulls  to  the  center  of  gravity  of  the 
recoiling  parts. 

In  recoil  systems  of  type  (2),  the  reaction  on 
the  carriage  consists  of: 

iii  a 

(1)  (pa ; )Ay  acting  to  the  rear 

"v 

(2)  n  Wr  +  RS+P  acting  forward. 

The  line  of  action  of  the  resultant  as  before 
passing  through  the  center  of  gravity  of  the  recoil- 
ing parts.  But  the  effect  of 

ilia  -,"a 

(pa  -    a )AV  is  exactly  the  same  as  Fv  -  ~ — 

wv  "v 

cV 

^«^ 
where  Fv  »  PaAv  and 


„ 

C0V 


"  y  ''y 

Hence  so  far  as  the  reactions  on  the  mount  and  motion 


469 


are  concerned,  the  hydro  pneumatic  and  spring  re- 
turn types  of  recuperators  are  exactly  similar. 

For  the  first  period  of  counter  recoil:  As- 
suming a  constant  orifice  during  the  first  part  of 

recoil,  we  have 

„ '  « 

C  v         dv 

Fv-Wr(sin0  +  u  cos  0)  -  Rs+p  -   '    =  mr  v  — 

"o        dx 

If  we  let,  R  =  (Wrsin  0+u  cos)+Rs+p  then 

C  v        dv 
Fy-R  -  —  =  mr  v  —        Now  Fy  is  a  function  of 

WQ       dx       x,  and  therefore  the 

equation  cannot  be 

readily  integrated.  But  since  Fv  does  not  vary 
greatly  over  a  short  interval,  we  may  assume  mean 
values  of  FV  for  a  few  given  intervals.   The  ad- 
vantage of  the  integration  of  the  equation,  is  that 
we  may  greatly  reduce  the  number  of  intervals  as  com- 
pared with  that  of  a  step  by  step  process  and  obtain 
sufficiently  exact  results. 

Considering  any  two  points  x±  and  xa  in  the 
counter  recoil,  we  have 

x         v    m_  v  dv 
/    dx  =  /    -— 2  where  A  =  Fy-R 

X  V       «     ^Ov 

xt        vt    A »--• 

wo 

Rearranging,  we  find     cv 

d(A~  ;T"' 

x  mw  *o 


hence 

2  2  a 

mug  f*  tr  /*  it 

v*  ™  *N  W   V  W   T 


and 


470 


*  *  * 

mrwo  wo  o 

therefore 

C0v*,  C0v*    2C(x,-xt) 

iog(A  -  -2-1) 


2.3mrwQ 


where  A  =  Fy-Wr(sin  0+u  eos  0)-R3<.p  (Ibs) 

Prom  this  equation  knowing  the  velocity  at  the 
beginning  of  any  arbitrary  interval  and  with  the  mean 
recuperator  reaction  for  the  interval  ire  may  compute 
the  velocity  at  the  end  of  the  interval.   Further 
fairly  large  intervals  may  be  assumed  provided  the 
recuperator  reaction  does  not  vary  greatly  at  the 
limits  of  the  interval. 

The  velocity  curve  is  computed  by  this  method 
to  x  =  bn-d  from  the  out  of  battery  position,  where 
d  =  length  of  the  retardation  or  variable  orifice 
period,  in  ft.  and  b  =  length  of  recoil  in  ft. 

The  velocity  v^  at  the  end  of  the  acceleration 
period  is  usually  taken  at  approximately  3.5  ft/sec, 
at  horizontal  elevation,  though  a  more  rational  as- 
sumption of  the  velocity  should  be  based  on  the  fol- 
lowing: 

Let  h  =  height  of  bore  from  ground  in  ft. 
(horizontal  c'recoil) 

d  *  length  of  buffer  or  variable  throttling 
interval  (ft) 

Ws  =  weight  of  total  system  gun  +  carriage  Qbs) 

18  =  horizontal  length  from  Ws  to  contact  of 

wheeled  ground  (ft) 

Cs  =  factor  of  stability  (»  0.85  usually) 
Rh  =  counter  recoil  reaction  at  horizontal 

elevation 

k  =  proportional  distance  of  d  that  the  c'recoil 
energy  is  to  be  dissipated  along,  k  =  0.7  to 
0.9 


471 


Then,  the  counter  recoil  reaction  at  horizontal  ele- 
vation, becomes, 


kd         h 

With  variable  recoil,  assuming  the  length  of  re- 
coil to  be  at  short  recoil  one  half  of  that  at  long 
recoil,  in  order  to  have  sufficient  displacement  for 
acceleration  at  maximum  elevation  the  buffer  or 
variable  throttling  should  not  take  place  at  horizon- 
tal recoil  for  over  1/3  to  1/4  the  recoil. 

Hence  d  «  0.33  to  0.25  b|,*0.3  bh  approx. 


/ 


°'6  C'"  *«1'">  -  3.62  /V£i.   (ft/,,0) 


Knowing  vb  we  may  estimate  the  proper  size  of  the 
counter  recoil  constant  orifice.  Actually  the  maximum 
velocity  of  counter  recoil  is  attained  shortly  after 
the  out  of  battery  position  and  at  this  position  the 

acceleration  is  zero.   But  since  the  retardation  is 
very  slight  until  the  variable  orifice  is  encountered, 
we  may  assume  the  recoiling  mass  to  move  with  uniform 
velocity  at  the  entrance  to  the  buffer  or  variable 
throttling.  Therefore  at  horizontal  recoil, 

=;•* 

P.  -  n  W_  -  ~ 


, 

W0 

Hence,  the  constant  orifice  becomes,  w( 


where  v^  *  3.62  /- where  for  a  spring  or 

"r  ^     pneumatic  return  re- 
cuperator system      (t  s 

i        b 

CQ   =    - 

175 

C  =  reciprocal  of  orifice  contraction  factor  and 
Afc  =  area  of  buffer  (sq.in) 

Fv  *  recuperator  reaction  at  displacement  X»bj,-d  (ft) 
and  for  hydro  pneumatic  recuperator  system, 


472 


-- 

C0  =     •        C  =  reciprocal  of  orifice  con- 

traction factor. 
Ay  =  effective  area  of  recuperator 

piston  (sq.in) 

FV  »  PaAv  lbs- 

p£  3  pressure  of  oil  in  air  cylinder. 

For  second  period  of  counter  recoil:   During  this 

period  it  is  customary  to  maintain  a  constant  total  re- 
tarding force  which  at  horizontal  elevation  becomes, 

cov* 
P-  n  1»- 


, 
where  Rh=cs"~h"  —  (lbs) 

Since  the  counter  recoil  reaction  is  constant 
during  the  retardation,  the  velocity  is  a  parabolic 
function  of  the  displacement,  that  is 


v  =  8.03  /  -          (ft/sec) 
"r 

Substituting  this  value  of  v  in  the  following  equation, 
we  have 


(sq.in) 

, 
b 


C  A 


where  for  a  spring  or  pneumatic  recuperator,  Co  =  • 

175 

C  *  reciprocal  of  orifice  contraction  factor 
Ak  =  area  of  buffer  (sq.in) 

2 

Fy  =  recuperator  reaction  at  displacement  x,   ^'  ^s 
for  a  hydro  pneumatic  recuperator  system,  C£  =  ""  • 

C  =  reciprocal  of  orifice  contraction  factor 
AT  =  effective  area  of  recuperator  piston 
Fv  =•  pi  Ay 

pa  *  pressure  in  oil  in  air  cylinder  (Ibs/sq.in) 


473 


COUNTER  RECOIL      With  a  variable  recoil,  the  re- 
FUNCTIONING  WITH  quirements  of  proper  counter  recoil 
VARIABLE  RECOIL,   functioning  for  all  elevations  are 
more  difficult  to  obtain.   At  hori- 
zontal recoil  we  must  meet  the  con- 
dition of  counter  recoil  stability,  whereas  at  maximum 
elevation,  the  time  period  of  the  counter  recoil,  for 
rapid  fire,  oust  not  be  too  long.   Since  the  recoil  at 
maximum  elevation  is  a  fraction  of  that  at  horizontal 
recoil,  the  recuperator  reaction  at  the  beginning  of 
counter  recoil  at  maximum  elevation  is  necessarily 
smaller  than  that  at  horizontal  elevation.   Further  at 
maximum  elevation  we  have  the  weight  component  resist- 
ing motion.  Therefore,  the  accelerating  force  is 
necessarily  considerably  smaller  than  at  horizontal 
elevation  and  the  velocity  attained  at  maximum  ele- 
vation becomes  a  function  of  that  at  horizontal  re- 
coil.  In  the  design  of  a  counter  recoil  system  in 
order  to  obtain  sufficient  velocity  in  the  counter  re- 
coil at  maximum  elevation,  it  is  important  that  a  proper 
compression  ratio  be  used.  This  in  turn  effects  the 
initial  volume  of  the  recuperator  and  therefore  the 
entire  layout  of  the  recuperator  forging.   It  is  here 
important  to  emphasize  that  proper  functioning  of 
counter  recoil  can  not  be  attained  by  increasing  pres- 
sure where  an  improper  ratio  of  compression  is  used. 

The  following  analysis  gives  a  rough  approximation 
as  to  the  requirements  to  be  met  for  proper  counter 
recoil  functioning  at  all  elevations  with  a  variable 
recoil. 

It  will  be  assumed  that  the  recoil  at  maximum 
elevation  is  reduced  to  one  half  that  at  horizontal 
recoil  and  that  a  constant  orifice  is  maintained  until 
the  latter  third  or  fourth  of  the  counter  recoil.  We 
have  therefore  a  constant  orifice  which  is  the  same 
for  the  accelerating  period  of  counter  recoil  at  max- 
imum elevation  or  horizontal  recoil. 


474 


If  now 

Fyj  =  initial  recuperator  reaction 

Fvf  =  final  recuperator  reaction   (Ibs) 

Fya)  =  recuperator  reaction  at  middle  of  hori- 
zontal or  long  recoil        (Ibs) 

vs   =  maximum  velocity  of  counter  recoil  at 
maximum  elevation  (ft/sec) 

vh  =  maximum  velocity  of  counter  recoil  at 
horizontal  elevation        (ft/sec) 
•Q  =  area  of  constant  orifice     (sq.in) 

Co  s  throttling  constant 

Rgifp  =  stuffing  +  packing  friction  (Ibs) 

0B  =  maximum  elevation 

As  a  first  approximation,  we  will  assume, 
the  maximum  horizontal  counter  recoil  velocity 
to  be  attained  after  a  displacement  equal  to  one 
half  the  recoil.  Hence 

cV 

Fvm  ~  n  Wr  -  Rs+p  -  —7 —  *  0  (1) 

"o 

At  maximum  elevation,  the  maximum  velocity 
of  counter  recoil  will  be  attained  somewhat  after 
a  displacement  equal  to  half  the  recoil,  but  we 
are  not  greatly  in  error  in  assuming  the  same  re- 
cuperator reaction  Fvin.   Hence 

<V's 
FVB-W   (sin0*n  cos   0)-  RS+D —  *   0  (2) 

*o 
Subtracting    (2)    from    (1).    we    have 

C0<vn-vs> 


W-[sin0-(l+  cos  0)nl  * 
i 


hence 

C0        Wr[sin0-n(l+  cos0)l 


475 


We  have  therefore  for  required  recuperator 
reaction  at  the  middle  of  the  recoil 

Wr[sin0-n(l+coslB)] 

If  we   assume   values   for     vh  and   vs   for  design  ap- 
proximations,   Me   may   take,vh   =   3.5   ft   per   sec,. 

vg   =   2.5   ft  per  sec. 
then,    FVJB=n  Wr+Rs+p+2Wr[  sin0-n(l+  cosfl)] 

If  we   take  a   large  coefficient   of  guide    frict 
ion  we   neglect  Rs+p;    hence   if   n   -  0.3, 
Fvm*0.3  Wr+2Wr[sin0B-0.3(l+  cos00)l 

To  obtain   the   minimum  allowable    ratio  of   compres- 
sion,   for   spring   recuperators,   we   have   2(?vm-Fyj)= 

F      )      hence 


Ffv   •    Fvi               FvfsFvi+< 

Fvf 

2(Fvm-0.5Fyi) 

m   : 

Fvi 

Fvi 

With  a   pne-umatic   or   hydropne-umatic   recuperat- 
or,   we   have   2.5  (Fvm-Fvi)=Fyf-Fvi      (approx.)     and 

Fyf=  Fyi+  2.5(F9m-¥vi)   hence 


2.5Fvm-   1.5Fvi        1.5(1.66Fvm-Fvi) 


=    n   = 


Fvi  Fvi  Fvi 

RKCUPBRATOBS. 

GENERAL  CONSIDERATIONS .     After  the  recoil  the 

recoiling  mass  must  be 
brought  into  battery 
and  this  must  take  place 
at  any  elevation  of  the 

gun  and  held  there  until  the  next  cycle  of  the 
firing.  Obviously  sufficient  potential  energy 
must  be  stored  during  the  recoil  to  overcome  the 

counter  recoil  friction  and  the  weight  com- 
ponent at  maximum  elevation  throughout  the  count- 


476 


er  recoil.   Further  in  order  that  the  counter  re- 
coil nay  be  made  in  mimimum  time,  an  excess 
potential  energy  is  required  over  that  required 
for  friction  and  gravity,  in  order  that  a  rapid 
acceleration  at  the  beginning  of  counter  recoil 
may  be  attained.   Finally  in  the  battery  position 
an  excess   recuperator  reaction  is  necessary  over 
that  for  balancing  the  weight  component  and  over- 
coming the  friction  in  case  of  a  slight  slipping 
back  of  the  piece  in  the  battery  position. 

Therefore  a  satisfactory  recuperator  must 
satisfy  the  following  requisites: 

(1)  The  initial  recuperator  react- 
ion should  have  a  marginal  excess 
over  that  requirad  to  balance  the 
friction  in  battery  and  the  weight 
component  at  maximum  elevation. 

(2)  The  potential  energy  of  the  re- 
cuperator at  the  end  of  recoil 
must  be  sufficient  to  overcome  the 
work  of  friction  and  gravity  at 
maximum  elevation  during  the  recoil 
and  rapidly  accelerate  the  gun  at 
the  beginning  of  counter  recoil. 

INITIAL  RECUPERATOR     In  general  the  size  or 
REACTION.  bulk  af  the  recuperator 

whether  spring  or  hydro 
pneumatic  depends  upon  the 
magnitude  of  t'he  initial  re- 
cuperator reaction.   It  becomes,  therefore,  im- 
portant to  estimate  the  required  initial  recuperat- 
or reaction  to  a  considerable  degree  of  accuracy. 
This  is  especially  true  in  certain  types  of  recoil 
systems  where  the  size  of  the  forging,  especially 
for  guns  of  high  elevation,  depends  directly  upon 
the  magnitude  of  the  initial  recuperator  reaction 

and  it  becomes  very  important  to  make  this  a  min- 
imum. 


477 


Let  Rg  =  guide  friction    (Ibs) 

R  v=  packing  friction  of  recuperator   (ibs) 
Fvi  =  initial  recuperator  reaction      (Ibs) 

ey  =  distance  down  from  center  of  gravity  of 
recoiling  parts  to  line  of  action  of  Pv 
(in) 

Qt  »  front  normal  clip  reaction    (Ibs) 
Qf  =  rear  normal  clip  reaction    (Ibs) 
x4  and  yt  -  coordinates  of  front  clip  reaction 

(in) 

xa  and  ya  =  coordinates  of  rear  clip  reaction 
(in) 

n  =  coefficient  of  guide  friction  =  0.15  approx, 

S6m-  angle  of  maximum  elevation. 

1  =  distance  between  clip  reactions  (in) 
Considering  the  recoiling  mass  at  maximum  elevation 
in  battery,  case  of  slight  slipping  back  from  the 
battery  position,  we  must  have  (see  fig     ) 


or 

Fyi=n(Q1+Q8)+Wrsin2fm  (1) 

and    normal   to   the  guides,    Qz-Qi=Wrcos(?  (2) 

and    taking    moments   about    the   center   of  gravity   of 

the   recoiling   parts, 

%i   *v-Qtxt-Q8xa  +  nQxyx-n  V,   =  °  (3  > 

Substituting  (2)  in  (3),  we  have, 

Fvi  ev~Qtxt~  Q2x8~wrcosgfx«  *  n  Qtyt"  n  a4y,  "  n 

Wf  cos/C  ya  =  0 

Fvi  ev-Wrcos0(xt+n  ya  ) 
hence  Qt  »  -   (Ibs)   (4) 

xt+Vn(y8-yt) 
and  solving  for  Q2, 


nQfy2    =  0 

v  a 

hence   CL   -  —  - 


Fvi 


478 
Hence,    with   sleeveguides, 

*_       "\y*— y* ' 

I  x  9  \ 

With  .grooved  guides  yt  becomes  negative  and 


Since  with  grooved  guides,  y^y^  approx.,also 
*!+*,  *  1  3  distance  between  clip  reactions,  and 

7tayt  *  er  »  lean  distance  to  guide  friction,  we 
have, 


v   vrBt 

Rff  »  -  n  (with  grooved  guides) 


2Fvi 


r,1a 

—  —  —  —  —  —  n  (with  sleeve  guides) 

1  (9) 


Substituting   in  eq.(l)      we    have 

t          2f'±  0_+ffrcoi0(x  -x..) 

-        —  -  -  S—  *-  n  + 


, 

1+2   n  e 

hence 


n  cos0_(x  -x    ) 
Wr[sineJm+  -  "      *      *    ] 

1+2   n  er 
-         (Ibs)      (10) 


2   e..n 


l+2n  er 

and  for  the  initial  recuperator  reaction, 

n  cos  £L(x  -x    ) 
'    3 


l+2n  e, 

^  n 


2  ev   n 
1       - — 


1+2   n  e, 


479 


n  cot  0B(xt-xt ) 
1+2  n  er 


2  ev  n 


>  where  k» 
1.1  to 


1.2    (11) 
1+2  n  er 

1  »  distance  between  clip  reactions  (in)  with  3 

clips  1  «-£ 

2 

with  4  clips:  1  »  b 

b  *  length  of  recoil  (in) 

Estimation  of  Recuperator  Packing  Friction  8p: 

With  hydro  pneumatic  recuperator  systems,  the 
packing  friction  is  usually  a  linear  function  of 
the  recuperator  pressure.   Assuming  a  given  in- 
itial intensity  of  pressure  pv  fflax  Ibs/sq.in.  in 
the  recuperator,  we  have,  Rp3Cppv  Bax. 

The  packing  friction  in  the  recuperator  is 
divided  into  the  suffing  box  friction  plus  the  re- 
cuperator piston  friction.   To  estimate  these  fric- 
tions wftmust  know  the  diameter  of  the  recuperator 
piston  rod  and  recuperator  piston. 

To  roughly  estimate  these  diameters,  we  have 
for  the  effective  area  of  the  recuperator  piston, 

1.3Hr(sin£5-+0.3  cos  0ffl) 
AT  -  (sq.in) 

PV  max 

for  the  required  area  of  the  recuperator  rod, 

2.6Wr(sin0m+0.3  cos  0m) 
av  « (sq.in) 

where  fB  »  allowable  fibre  stress  in  rod  material. 
Then  the  diameter  of  the  piston,  becomes, 

(in) 
0.7854 


480 


and  the  diameter  of  the  rod  becomes,  d  ' 

v  '0.7854 

(in) 
If  wgv  =  width  of  stuffing  box  packing  of  recuperat- 

or(assumed ) (in) 
wpv  *  width  of  piston  packing  of  recuperator 

(assumed  Kin  ) 

then  assuming  the  pressure  normal  to  the  cylinder 
or  surface  of  the  rod  to  be  made  equal  to  the  hy- 
drostatic pressure  in  the  cylinder,  we  have 
Rp-(.06«  wpv  Dv  +  .05  n  w3v  dy)py  nax< 

-  .05n(wpy  Dv  +  wsy  dv)py  max!  (Ibs) 

where  .05  »  approx.  coefficient  of  friction  of  the 
packing. 

Approximate  Initial  Recuperator  Reaction: 

For  preliminary  calculations,  especially  when 
the  type  of  packing  and  arrangement  of  cylinders 
has  not  been  considered  we  may  neglect  the  re- 
cuperator packing  friction  by  increasing  the  co- 
efficient of  guide  friction. 

Without  pinching  action  of  the  guides  in  bat- 
tery the  guide  friction,  R-  »  0.15  Wrcos  0  (approx) 
(Ibs).  To  account,  for  a  possible  pinching  action, 
as  well  as  the  packing  friction,  for  elevations 
up  to  65°,  approx.  Rg  =  0.30  Wr  cos  0   (Ibs)  and 
the  required  initial  recuperator  reaction,  to  al- 
low for  possible  variations,  should  be  increased 
from  20*  to  30*  over  that  required  to  hold  the  gun 
in  battery.   Hence  Fyi  *  1.3  Wr(sin0m+  0.3  cos£5) 
(Ibs).   With  guns  of  very  high  elevation,  Rg  = 
0.3  cos  t  becomes  negligible.   However,  the  pack- 
ing friction  remains  the  same  whereas  the  guide 
friction  is  comparable  with  that  at  horizontal  re- 
coil due  to  the  pinching  action  of  the  guides  at 
maximum  elevation.   Therefore,  it  is  desirable  to 
use  an  approximate  formula  taking  these  factors 


481 


2n 
into  consideration.      We   have,    appro*.  Re   = 

1 

where  n  =  0.1  to  0.2.   If  we  take  n  *  0.3  to  ac 
count  for  the  recuperator  packing  friction,  we 
have  at  high  elevations, 

0.6  Fvi  eb 
pvi  -  1.3(Wr  sin  0m  +  *Y* )    (Ibs) 

where  eb  *  distance  from  bore  to  line  of  action 

of  Fvj  (assumed ) (in) 

1  *  distance  between  clip  reactions  (in) 
with  3  clips  


with  4  clips 1  =  b 

b  -  length  of  recoil   (in) 

ENERGY  REQUIREMENTS     The  initial  recuperator 
FOR  PROPER  reaction  is  designed  to  be 

RECUPERATION.         somewhat  greater  than  that 

required  to  hold  the  gun  in 
battery  at  maximum  elevat- 
ion, against  the  guide  and  packing  frictions. 
Further,  the  recuperator  reaction,  being  necessarily 
derived  from  a  potential  function,  must  therefore 
increase  with  the  displacement  out  of  battery. 
The  work  done  by  the  recuperator,  therefore,  is  in 
excess  of  that  required  and  we  have,  always,  an 
excess  potential  energy  over  that  required  to  bring 
the  gun  into  battery.  This  excess  energy  is  dis- 
sipated by  the  counter  recoil  regulator.   We  have, 
i 

therefore,  merely  a  transfer  of  part  of  the  re- 
coil energy,  dissipated  by  Beans  of  the  recuperat- 
or, ultimately  in  the  counter  recoil.   The  total 
heating  or  rather  the  average  in  a  recoil  cycle 
is  quite  independent  of  the  magnitude  of  the  com- 
pression.  However,  with  high  compression  ratios, 
we  have  extreme  local  heating  where  the  radiation 
is  small  and  therefore  injurious  effects  are  like- 
ly to  result  with  the  air  packings  in  hydro  pneu- 


482 


•atic  recoil  systems.  Further  excessive  potential 
energy  stored  in  the  recuperator,  requires  care- 
ful counter  recoil  regulation,  and  as  stability 
on  counter  recoil  is  far  more  sensitive  than  on 
recoil,  we  have  more  difficulty  in  meeting  the 
rigid  requirements  of  counter  recoil  stability. 
Finally  with  excessive  recuperator  energy  to  main- 
tain low  counter  recoil  regulator  or  buffer  pres- 
sures requires  a  cumbersome  and  large  counter  re- 
coil regulator  whereas  it  is  far  simpler  con- 
structively to  dissipate  the  recoil  energy  during 
the  recoil. 

Therefore  excessive  recuperator  energy  is  un- 
desirable for  the  following  reasons: 

(1)  Localized  heating  resulting  with 
hydro  pneumatic  recuperators,  is  in- 
jurious to  the  packing. 

(2)  Difficulty  in  counter  recoil 
regulation  and  meeting  counter  re- 
coil stability  requirements. 

(3)  Constructive  difficulties  due 
to  a  bulky  counter  recoil  buffer  or 
regulator  required  to  maintain 
moderate  pressures  in  the  buffer 
chamber. 

On  the  other  hand,  the  mean  recuperator  re- 
action must  be  sufficient  not  only  to  balance 
the  weight  component  of  the  recoiling  parts  and 
frictions,  but  enough  to  accelerate  the  recoiling 
parts  to  a  given  minimum  velocity  for  counter  re- 
coil at  all  angles  of  elevation.   Since  it  is  con- 
structively complicated  and  more  or  less  impractical 
to  introduce  varying  counter  recoil  regulation  as 
the  gun  elevates  in  the  majority  of  the  types  of 
recoil  systens  are  designed  on  the  bases  of  given 
maximum  velocity  at  horizontal  elevation  consist- 
ent with  counter  recoil  stability  and  a  given 
minimum  velocity  at  maximum  elevation,  consistent 
with  reasonable  time  of  counter  recoil  at  maximum 


483 


elevation.  Usually  the  recoil  is  shortened  at 
•aximum  elevation.  We  are  not  greatly  in  error 
in  assuming  the  respective  velocities  to  be  at- 
tained at  a  displacement  corresponding  to  the 
oean  recuperator  reaction,  whicb  is  roughly  from 
one  half  to  two  thirds  away  from  the  battery 
position. 

We  have,  then,  with  a  variable  recoil,  if 
Pvm  =  mean  recuperator  reaction   (Ibs) 
Rs+p  =  total  packing  friction  in  counter  re- 
coil  (Ibs) 

CQ  =  throttling  constant  of  regulator 
w0  »  throttling  orifice  of  regulator  (sq.in) 
vb  »  velocity  of  horizontal  e 'recoil  (ft/sec) 
vs  =  velocity  of  c'recoil  at  maximum  elevation 

(ft/sec) 
n  -  coefficient  of  guide  friction, 

for  the  notion  of  the  recoiling  parts  at  horizont- 
al recoil, 

<*; 

%«  -  "  »r  -  Rs+p  -  —  =  0 

for   the   motion  of   the  recoiling   parts   at   maximum 

elevation, 

_  i    * 
C0vs 

FVB  -Wr(sinJ0m+n  cos  0m)+Rs+p jj—  »  fr 

wo 
Subtracting,    we  obtain 

C0         Wrtsin0m-  n(l 


2 

vb 


and 

X 

tfh 

FVIB*   n 

wr+Rs+p  + 

W_[sin 
y»    _   va    r 
vh       vs 

cos 


We  see,  therefore,  that  the  mean  recuperator  re- 
action required  depends  greatly  on  the  square  of 
the  horizontal  c'recoil  velocity  and  inversely  as 
the  difference  between  the  squares  of  the  borizont- 


484 


al  and  maximum  elevation,  o  'recoil  velocities. 
Since  vn  is  nore  or  less  fixed  by  c  'recoil  stability 
limitations,  whereas  vg  depends  upon  the  time  allowed 
for  counter  recoil  functioning  at  maximum  elevation, 
Fym  becomes  more  or  less  fixed  and  therefore  the 
required  excess  potential  energy  of  the  recuperat- 
or. 

Assuming  design  values  of  vh  =  3.5  ft/sec. 
and  vs  =  2.5  ft/sec,  with  an  increased  coefficient 
of  guide  friction  to  compensate  for  the  packing 
friction,  n  =  0.3,  we  have 

Fym  =  0.3Wr+2Wr[sin0m-0.3(l+cos  0m)]  which  gives 
a  rough  approximation  as  to  the  value  of  the  mean 
recuperator  reactions  required. 

CALCULATION  OP  THE  MEAN  RECUPERATOR  REACTION 
AND  THE  BNERQY  STORED  IN  THE  HBOUPBHATOR. 

SPRING  RECUPERATORS.         With  spring  return 

recuperators,  we  have 
the  recuperator  re- 
action increasing  pro- 
portionally with  the 
recoil.   If  Fvi  *  SQ  =  the  initial  spring  re- 

cuperator reaction  (Ibs) 

Fvf  *  sf  =  the  final  spring  recuperator  re- 
action (Ibs) 


b  -  length  of  recoil  (ft) 
Then       Sf+So   Fyi*Fvf 


(lbs) 


hence  Tvf   *  2Fym-Fvi      (Ibs) 


The  potential  energy  stored  in  the  recuperat- 
or for  s  displacement  x,  becomes 

x  Sf-S0 

W   »  /      (S0+  —  -  —  x)dx 


485 
c  _—  c 

bf      bO         2 

=   S0   x   +  -  x      (ft.lbs) 
2b 

and   the    total  potential   energy   required   at   the   end 
of   recoil,   becomes 

W   =    (S0   +   Sf)|  -    (Pyi   *   Fvf)|      (ft.lbs) 

With  hydro  pneumatic  or  pneumatic  recuperat- 
ors, we  have  the  recuperator  reaction  increasing 
as  an  exponential  function  of  the  recoil  displace- 
ment.  If 

pa  a  intensity  of  air  pressure  in  recuperator 
at  any  displacement  in  the  recoil  X 
(Ibs/sq.ft) 

*  initial  pressure  in  the  recuperator 

(Ibs/sq.ft) 

*  final  or  maximum  pressure  in  the  re- 
cuperator  (Ibs/sq.ft) 


•  »  ^.   -  ratio  of  compression 
Pai 

Ay  *  effective  area  of  recuperator  piston  (sq. 

in) 
V  *  volume  of  recuperator  at  displacement  x 

(cu.ft) 

VQ  =  initial  volume  of  recuperator  (cu.ft) 
Vf  =  final  volume  of  recuperator  (cu.ft) 
x  =  recoil  displacement  (ft) 
b  =  total  length  of  recoil  (ft) 
Then, 

Pa^  a  Pai  vo 
where  k  3  1.1  for  oil  in  contact  with  air 

=  1.2  for  oil  separated  from  air  by  a  float- 

ing piston. 

Since  V  =  VQ  -  Ayx,  for  a  recoil  displacement  x, 
we  have         y     ^ 

pa  =  pai  (tr-^T  -  ^     or  ln  ternis  of  tne 

°  v        total  recuperator  re- 
action 


486 

r 


0 


O   T 

Tba  work  of  coapression,  becoaes 

V  V  dv 

"x  »  -  /  P.  d  7  »  -  pai  Vj  /   21   tft.lbs) 

V  V   Vk 


1-k 


Since  Fy±  =  PaiAv,  we  have  for  the  work  of  com- 
pression in  terms  of  the  total  initial  recuperat- 
or reaction 

Fvi  Vo   ,   1       1   . 

wx  '  I- 1  k-1)  ^yk-i      v*-1 

*o 

where  as  before  V  =  VQ-Ayx.   At  the  end  of  recoil, 
we  have  substituting,  for  V,  7f  = 


•  »  —  —  *  £77-)  »  the  ratio  of  compression. 


now 


Tbe  total  work  of  eoapression  in  terns  of  "a"  be- 
comes     -   y   *;i 

Wb  -  -2 S(H  k  -  1)  (ft. Ibs) 

k-1 

It  is  custoaary  to  measure  the  pressure  in 
Ibs.  per  sq.in.  rather  than  Ibs.  per  sq.  ft.  and 
the  volume  in  cu.  in.   The  above  formulas,  be- 
come 

Pai  7o      I        1 


12C1C-1J 


487 


-V      — 

(B    -  1)  (ft.lbs)  or  in  terns  of 


the  initial  re- 

cuperator reaction  Fvi  and  the  effective  area  of 
the  recuperator  piston  Av(sq.in)  we  have 


12Av(k-l) 


(ft.lbs) 


P  -  V*    ~ 

wb  =  —  —  —  2—  (•  '  -  i)  (ft.  ibs) 

12Av(k-l) 

v 

and  PaAv  =  Fv  =  pai  Av(  -  -  -  J  (Ibs) 

W 
where  x  =  recoil  displacement   (inches) 

Ay=  effective  area  of  recuperator  pistomsq  .in; 

Vo=  initial  volume  (cu.in) 

paj=initial  recuperator  pressure  (Ibs/sq.in) 

V  =  V0-Avx   (cu.in) 
The  mean  recuperator  reaction,  becomes, 

Fvivo     *** 

^m     -  1  )  (Ibs)  where  Ay  is  in  sq. 

>  ft.,  b  in  ft.,  and 

Paf       V  Vo  in  Cu'ft' 

Since  -  »  m  =  (--) 

Pai        Vf 


and  V*  •«  —     hence 


1  m   -  1 

VQ(1  --  )Avb   and  Ayb  »  VQ(  -  )  therefore 

m*  at* 

1        k-l 

Fv«  *  Fvi(~  -  H*   "1)  (Ibs)  which  gives  the 

mean  recuperator 
reaction  in  terms 

of  the  initial  recuperator  reaction  and  the  ratio 

of  compression 


488 


Since  Pvj  *  1.3yfr(sin0a+0.3  cos  0m)(approx.) 
(Ibs)  we  will  have 


—      0.3Wr(  ,   t)Wr[3in0B-0.3(l-cosg),)] 


k-1 

1.3Wr(sin0B+0.3  cos0m) 


1.3(8in0B+0.3cos0B) 

If  we  assume  vn  -  3.5  ft/sec,  and  vs  =  2.5  ft/aec,, 
then 


hence 
1_      Jt=i 

0.3+2[sin0m-0.3(l-cos0B)] 


^T"  -)(n^r^ 
•*  - 1 

From  the  above  equations,  we  note  that  the 
proper  ratio  of  compression  depends  on  the  angle  of 
elevation  and  is  entirely  independent  of  the  weight 
of  the  recoiling  parts.   The  compression  ratio 
does  depend  upon  the  value  assumed  for  the  initial 
recuperator  reaction,  the  higher  the  initial  re- 
cuperator reaction  the  lower  ratio  of  compression. 
The  compression  ratio  increases  with  the  elevation 
for  proper  functioning  of  counter  recoil  at  max. 
elevation. 

If  now  we  construct  a  table  with  values  of  m, 
and  the  corresponding  values 


489 


( )      and       a ~    and   their  product   for 

1  k-1 

k  =  1.1  and  1.3  res- 
pectively, we  >ay  de- 
termine •  by  inspection  and  interpolation,  pro- 
vided we  know  the  max.  angle  of  elevation.   If 
we  let, 

1  k-i 

B*  -  *  -  1 

A  -  - ;   B  • 


k-1 


r=-T)  [  s  i  n0m-0 . 3  ( 1-c  os£fB )  ] 


then,   where   k  =1.1 
•  A 


1.3 

4.71V 

o.  23 

1.O84 

1.5 

3.24'7 

0.37 

1.201 

1.75 

2.  §08 

0.51 

1.279 

2.00 

2.  138 

0.  64 

1.368 

2.  3O 

1.  883 

0.  78 

1.  468 

and  where   k  -   1.3 

m  A 


1.3 

5.464 

.296 

1.  129 

1.5 

3.732 

.326 

1.  219 

1.75 

2.  840 

.456 

1.306 

2.0O 

2.  420 

.577 

1.395 

2.  30 

2.  113 

.703 

1.  486 

490 


from  the  above  tables,  carves  were  plotted  with 
values  of  C  against  •  for  k  *  1.1  and  1.3  respect- 
ively. 

la  order  to  compare  the  probable  velocities 
obtained  in  the  counter  reeeil  at  maximum  and  hori- 
zontal recoil  for  a  given  ratio  of  coapression  m, 
or  on  the  other  hand  if  given  values  of  velocity 
at  borisontal  and  naximua  elevation  are  wanted  the 
following  method  enables  us  to  determine  the  proper 
value  ef  the  ratio  of  compression  •. 

If  we  plot  for  various  values  of  m,  the  cor- 


responding value  of 


for  a  lean  aax.  elevation  rg  at  63°,  against  vh  as 
horizontal  abscissa  and  vs  as  ordinates,  we  obtain, 
a  series  of  curves  for  the  various  values  of  a, 
which  having  decided  upon  the  ratio  of  compression 
to  be  used  enables  us  to  determine  immediately  the 
velocity  of  c  'recoil  at  max.  elevation  for  any 
given  velocity  at  horizontal  recoil. 
How,         ^ 

0.3+  -  -  —  [sin*H5.3(l-  cos  £J)] 


1.3(sin0+0.3  cos  0) 

0  *  angle  of  elevation.   (In  this  series  of  cal- 

culations, the  angle  of  elevation  will  be  con- 

sidered only  at  65°). 
.-.  0  -  65°. 

Sin  t  -  .906308 

Cos  0  *  .422618 

Sin  t  -  0.3(l-cos0)»  .906308  -  .3(1  -  .422618)*  .7330 
1.3(sin0+.3cosO)»  1.3(.  906308+  .3  x  .422618)«1.343 

Various  values  ef  C  (equation  fl)  are  given 
in  table  on  preceding  page. 

The  only  unknown  in  the  equation  111  is  the  ex- 
pression 


491 


Vu  « 

vh  y" 

•   Let        =  K.   Taking  the  various  values 

Vn~"Vs  of  C  as  given  in  the  pre- 

ceding table  and  sub- 
stituting in  formula  #1,  we  get  the  following  values 
of  "K",  for  the  given  values  of  "C": 


1.1 


1.084 

1.5*76 

1.  201 

1.790 

1.2-79 

1.933 

1.368 

2.096 

1.  468 

2.279 

1.3 


1.129  1.659 

1.219  1.824 

1.306  1.983 

1.395  2.145 

1.486  2.312 

..-!L 

v«-va 

h  s 

Now  to  show  the  relation  Vh  and  Vs,  a  curve 
will  be  plotted  for  each  value  of  "K"  as  calculated 
and  recorded  in  the  table  : 

Vh  -  KVn  -RVs   ;  K7!  =  EV*h  -  ?J  ; 


492 


KV'U-V! 


V. 


P 
T 

Now  for  each  value  of  K,  assume  values  of  Vh,  from 
0  to  10  and  substitute  in  Formula  #2,  and  obtain  various 
corresponding  values  of  Vs.  These  values  of  Vs 
plotted  against  values  of  Vn  enables  us  to  plot 
the  curve,  the  corresponding  values  of  V8  and  Vh 
for  each  value  of  "K". 


1.1     SET    Or    CURVES 

M 

When    K  -   1.576  1.3 

Vh         123*5678  10 

Vg     .604     1.21    1.81     2.42     2.79     3*62     4.08    4.84        6.05 

When  K  -  1.790  1.5 

^h        12345          673          10 
V,    .663    1.33    1*98    2.66   3.32    3.96    4.64   5.31    6.64 

When   K  -   1.933  1.75 

Vn        12345  6*78  10 

V3     .693    1.39    2*08    2.78    3*47    4.16    4.85    5*55    6.93 

(Then   K  »   2.096 
Vh        123*5678          10 

Vg    .72.2     1.44     2.17    2.89    3-62     4.34    5.06    5.78    7.22 

When   K  -   2.279 
Vb        12345          67s          10 

V,     .748     1.50     2.24     2.99     3.74     4.48     5.24     5.99     7.49 


493 


1.3   BIT   or   CURVES. 


When   K  »   1.659  1.3 

Vn  12345678  10 

Vg        .63        1.28     1.89     2.52     3.15     3.78     4.41    5.04     6.3 

K  -   1.824  1.5 

Vh  12345678  10 

V_       .67       1.34    2.01    2.69    3.36    4.03    4. "70    5.37    7.12 

• 

K   »    1.983  1.75 

Vh  12345678  10 

Vg         .704     1.40     2.06     2.82     3. 52     4.22     4.92     5.63     7.04 

K   »   2.145  2.00 

Vh          1-2345678          10 

V8      .734    1.46   2.19    2.92    3.65    4.38    5.11    5.35    7.30 

K  -   2.312  2.3 

Vh          12345678          10 

V3       .753    1.50    2.26    3.01    3.76    4.50    5.27    6.02    7.53 


SPRING  RECUPERATORS.     Spiral  spring  columns,  en- 
closed in  cylinders  for  pro- 
tection, are  extensively  used 
to  bring  the  recoiling  parts 
back  into  battery  from  the  out 

of  battery  position.    For  small  guns,  spring  re- 
cuperators are  more  useful,  since  they  are  simple 
in  construction  compact  and  readily  adaptable  to 
a  gun  mount.   With  large  guns,  however,  the 
energy  required  for  recuperation  is  large  and  there- 
fore the  spring  columns  become  excessively  heavy, 
since  the  weight  of  the  springs  is  proportional  to 
the  potential  energy  stored  within  the  springs. 


494 


495 


496 


497 


Hence  for  large  guns  pneumatic  recuperators  have 
become  almost  universally  employed. 

The  stresses  computed  in  springs  are  based 
merely  on  their  static  loading.   During  the  ac- 
celeration period  of  the  gun,  the  spring  coils 
adjacent  to  the  attachment  on  the  recoiling  parts, 
necessarily  are  subjected  to  a  very  large  ac- 
celeration, whereas  those  coils  adjacent  to  their 
attachment  on  the  cradle  remain  stationary.   Due 
to  the  great  resilience  of  a  spring  column,  probably 
only  a  few  of  the  front  coils  adjacent  to  the  re- 
coiling parts  are  subjected  to  any  material  accel- 
eration, the  spring  not  being  capable  of  transmit- 
ting a  force  sufficient  to  accelerate  the  inner 
coils.   Due  to  the  very  rapid  acceleration  during 
the  first  part  of  the  powder  period  we  have  an  im- 
pact or  very  suddenly  applied  loading  on  the  spring 
which  induces  a  compression  wave,  the  peak  of  the 
wave  being  adjacent  to  the  recoiling  parts  and  the 
velocity  of  which  depends  upon  the  inertia  per  unit 
and  elastic  constant  of  the  spring.   It  is  possible 
that  some  of  the  failures  in  the  service  of  re- 
cuperator springs  are  due  to  the  dynamical  aspects 
of  the  loading  on  the  springs  during  the  firing. 

Since  the  inertia  loading  due  to  the  powder 
acceleration  comes  practically  on  the  front  series 
of  coils  adjacent  to  the  recoiling  parts,  the  coils 
directly  adjacent  to  the  recoiling  parts  become 
more  greatly  compressed  and  correspondingly  stressed. 
We  should  expect  the  front  coils,  therefore,  to 
give  the  greatest  trouble  and  this  has  been  found 
the  case  in  actual  service. 

Due  to  the  complexity  of  the  problem  in  actual 
calculations  of  the  dynamic  stresses  in  the  spring 
no  attempt  will  be  made  here  to  outline  a  procedure 
for  such  calculations,  and  only  the  static  loading 
with  suitable  safety  factors  based  on  experience 
will  be  used  in  the  preliminary  design  of  counter 


498 


recoil  springs.    Lei 

0  =  diam.  of  the  helix  of  the  coiled  spring 

(in) 
R  *  radius  of  the  helix  of  the  coiled  spring 

(in) 

d  =  diam.  of  the  wire  (in) 
fs=  max.  allowable  torsional  fibre  stress 

used  (Ibs/sq.in) 
N  3  torsional  modulus  of  elasticity  (Ibs/sq. 

in) 
T  =  torque  or  total  torsion  at  any  cross 

section  of  the  wire  (in.  Ibs) 
Considering  any  portion  of  a  spring  column 
subjected  to  a  conpressive  load  F  (Ibs)  ,  along 
the  helical  axis,  we  have  at  any  section,  through 
the  wire, 

A  torsional  load  T  =  f  R 
A  shear          S  =  P 

If  we  assume  pure  torsion  at  the  section,  the 
torsional  fibre  stress  becoaes, 

f  »  fg  —   (Ibs/sq.in)  where  ro  =  -   (in)  hence 
ro 

T  »  F  R  -  /  °  2*r  dr  f.  — 

s   T* 

o  ro 


and  therefore         ,3        a 
nfad    n  fsd 

F  »  =  (Ibs) 

16R      8D 

Next  consider  the  twist  of  any  length  of  the 
wire  1.   We  have,  for  the  torsional  shear  displace- 
ment of  a  circumferential  annular  of  the  wire, 

fs  ro 

t  *  —  since  fa  -  0  N  hence  0  »  r-  where  9  «  the 
N  1 


499 


angle  between  two  radius  of  the  wire  at  two  sections 
1  distance  apart.   Therefore 

a  ,£^«!£ii 

The  relative  displacement  between  the  extremities 
of  the  helix  lor  a  load  P,  producing  an  extreme 
fibre  stress  f,  becomes 

2f,Rl 

9  »  R  9  »  — but  the  length  of  the  total  wire 

Nd 

of  the  helix,  becomes,  1  »  2nRa 

approx.  *  K  Dn  where  n  -  no.  of  coils.   Bonce 
9 

We  have,  therefore,  the  two  fundamental  spring 
formulas,  for  springs  of  circular  cross  section 

Kf  d*    ufsds 

PR =  (Ibs)          (1) 

8D        16R 

nf.Dfn 
9  »  — (in)  (2) 

B  d 

The  above  formulas  apply  strictly  only  to 
closed  coiled  springs,  no  bending  being  considered; 
however,  for  a  first  approximation,  they  may  be 
used  for  open  coiled  springs  with  sufficient  accuracy 
for  ordinary  calculations. 

For  rectangular  wire  sections,  we  have  semi- 
empiroal  formulas  for  the  torsion,  and  deflections; 

*  *   <«!*?  PK    >f» 

9 


.8b 
4  o  J  T  1 


where  aQ  *  length  of  long  side  of  rectangular  section 

(in) 
bQ  =  length  of  short  side  (in) 


500 


J  a  the  polar  moment  of  inertia  of  the 

rectangle. 

1  *  length  of  wire  (in) 
A  *  cross  section  of  the  wire  (sq.in) 
No"  ab*    ba3    ab(a*+b*) 

lxx+lyym  12"   "12  3     12 

Hence  for  rectangular  section  spiral  springs, 
we  have, 


aobo 

fs 

~(     aobo 

}   ', 

3a0+1.8b0 

1 

3a0+1.8b0 

D 

lOnJD'n 

—  .  p 
A4    N 

166nD'n        a0b0(a0+b0) 

-  --  ta   (in)    (2») 
A*N     (3a0+1.8b0) 

If  now,  we  let 

a  =  deflection  at  assembled  or  battery  height 

(in) 
b"=  displacement  somewhat  greater  than  the 

length  of  recoil  (in) 
Fyi  «  load  at  assembled  height  )  initial  re- 

cuperator reaction  (Ibs) 
Fvfl  *  load  at  solid  height  or  at  deflection 

corresponding  to  (a+b  '  ) 
n  *  no.  of  effective  coils 
N  »  torsional  modulus  of  elasticity  (Ibs/sq. 

in) 

d  *  diaa.  of  wire  (in) 
D  *  diai.  of  helix  (in) 
H0*  solid  height  of  spring  (in) 
f3=  working  «ax.  fibre  stress  (Ibs/sq.in) 
then: 

for  circular  springs:    for  rectangular  springs: 


ab  f 


501 


1.66nD*n 


A4  tf   "  (3aQ*1.8b0fs 
(in)  (2') 


vi 

~  ' 


H0  -  nd      (4) 

In  the  four  equations,  above  we  are  given  fg 
P?9  bf  N  and  Ho 

Fve 
fa  ?„*  b '  N  D  and  - — 


leaving  the  four  unknowns,  d  D  a  and  n  or  d  a  n  and 

H0.  Therefore  a  complete  solution  is  possible,  and 
the  proper  size  spring  may  be  iauaed lately  arrived 

at. 

ENERGY  STORED  IN  SPRING     The  fibre  stress  on  a 

helical  spring  is  direct- 
ly proportional  to  the 
axial  load,  that  is 

f  *  — TS-  F  (Ibs/sq.in)  and  the  corresponding  axial 
n  u  _  ns  g 

deflection,  becomes,  9  *     f  (in)  hence  the  de- 

Nd        flection  of  a 

helical  spring  loaded  axially  is  directly  pro- 
portional to  the  load,  that  is 

6  = f   (in) 

Hd4 

The  potential  or  resilient  energy  stored  in  a 
helical  spring  becomes, 

P  „    Hd4  at   ,. 

A  »  -  9  =  — . —  9    (in  Ibs) 

2     16D»n 

If  the  spring  is  to  be  stressed  to  a  maximum 
allowable  fibre  stress  fs  (Ibs/sq.in)  we  have 


502 


n'Dd'n 
A  »   16  N  f*8  (in  Ibs) 

The  volume  of  the  material  of  the  spring  equals 
approximately 

n   »         n    a 
7  »  -  d  n  *  D  »  -  D  d  n  (cu.in) 
4  4 

Hence  the  total  energy  in  terns  of  the  volume,  is 

»f*s 

A  »  -  —  that  is,  the  energy  stored  in  a  spring 
4   N 

for  a  given  max.  allowable  fibre  stress 

and  torsional  modulus,  is  directly  proportional  to 
the  volume  and  hence  the  weight  of  the  spring.  Thus 
with  tbe  same  maximum  stresses  and  same  kind  of 
material,  the  weight  of  the  spring  is  directly  pro- 
portional to  tbe  energy  absorbed  by  tbe  spring* 

The  weight  of  tbe  spring  in  terms  of  the  total 
energy  stored  in  the  spring,  becomes 

where  Wg  *  total  weight  of  tbe  spring 

(Ibs) 

ws  =  weight  per  cu.in.  of  tbe 
material  of  tbe  spring 
(Ibs/cu.in) 

RATIO  OP  COMPRESSION  WITH     For  minimum  weight 
SPRING  RECUPERATORS  FOR    of  a  set  of  counter  re- 
MINIMUM  WEIGHT  OP  COUNT-   coil  springs  the  com- 
EB  RECOIL  SPRINGS.         pression  ratio  is 

definitely  fixed. 

Let  Fvi  *  the  initial  recuperator  reaction  (Ibs) 
Fvf  =  tbe  final  recuperator  reaction  (Ibs) 
Pye  =  the  maximum  solid  load  on  the  re- 
cuperator springs  (Ibs) 

a  *  deflection  of  springs  to  assembled 

height  in  battery  (in) 
b  *  lengtb  of  recoil  (in) 
b"  *  detleotion  of  springs  from  assembled  to 

solid  height  (in) 


503 


Since  the  load  on  the  springs  is  proportional 
to  the  deflection  we  have  immediately, 

Fre   a+b*  b" 

-  *  -  ;  and  Pra  -  Fvi(1+  —  > 


a 


The  total  energy  stored  in  the  spring  column, 


A  -  -  (a+b")   =  -  U+2b'  +  -  —  )(in.lbs) 
222 

Since  b"   and   Pyi  are   fixed  conditions   to  be   not 
in  the   design  of   the   carriage,    the   only  variable 
in  the   above   energy  expression   is   a.     Therefore, 

for   minimum  weight,  w  «a 

d(a+2b"+  -  ) 

dA  a 

—  s  0    hence 


da  da 

1  *  —5—  »  0  and  therefore  a  »  b" 

a 

The  ratio  of  compression,  becomes, 
— — •  »  2  »  —  (approx) 

Fortunately  this  ratio  is  nearly  ideal  for 
proper  recuperation  and  hence  satisfactory  de- 
signed spring  column  with  minimum  weight  may  be 
used . 

RECOPERATOR  DIMENSIONS         With  hydro 
AND  LIMITATIONS.        pneumatic  recuperators 

we  have  two  or  more 
cylinders,  the  recuperat- 
or cylinder  and  the  air 
tank  or  cylinder.  Let 

b  =»  length  of  recoil  (in) 

b  =  corresponding  displacement  in  air  cylind- 
er 


504 


Ay  =  effective  area  of  recuperator  piston  (sq. 

in) 
Aa  »  cross  section  area  of  air  cylinder  (sq. 

in) 

Paf 
m  »  *  ratio  of  compression 

Pai 

Aa 
r  *  — -  *  ratio  of  recuperator  cylinders. 

Ay 

1  =  length  of  air  volume  in  terms  of  cross 
section  area  of  air  cylinder  (in) 

j  »  -  3  length  of  air  volume  in  terns  of  re- 
coil stroke 

Vo  *  initial  air  volume  (cu.in) 
Vf  *  final  air  volume  (cu.in) 
Then  Vf  «  VQ-Avb 


U  U  V 

P 

VQ  k 

-i£  « 

m  *    (—  )        where 

k  *    1.1   to  1.3 

Pai 

vf 

therefore. 

vo 

i 

vo 

*r  3 

• 

that    is   Vf  « 

"T 

mk             1 

1 

, 

k 

_k 

VI 

1)= 

>ATb          hence 

V-                                  A      V,      • 

m 

0    "        ,              AVb    * 

i 

'Aab' 


•"  mk-  1       mk  -1 

which  shows  clearly,  that  the  initial  volume  de- 
pends only  upon  the  ratio  of  compression,  the 
area  of  the  recuperator  cylinder  and  the  length 
of  recoil. 

If  now,  we  decrease  the  effective  area  of  the 
recuperator  piston,  for  a  given  recuperator  re- 
action, we  must  increase  the  intensity  of  pres- 
sure in  the  recuperator  cylinder,  that  is: 

•r  M       Y\  A 

fty|  Pyi       fly 


505 


Kvib  n 
hence  VQ  *  — —   1--        (6) 


since  pvi  »  pal  »  pai  approx, 

i 

Kvib    B= 
V0  -  -      i  (61) 

Pal   m*  -  1 

Now  the  size  of  the  recuperator  depends  rough- 
ly on  the  initial  volume  VA;  hence,  in  pneumatic 
or  hydro  pneumatic  systems,  it  is  important  to  main- 
tain as  high  air  pressure  as  possible. 

In  recoil  systems,  where  the  recuperator  and 
brake  cylinder  is  one  and  the  same  as  in  the  St. 
Chamond  and  Puteaux  brakes,  the  effective  area  of 
the  recuperator  piston  is  that  of  the  recoil  pis- 
ton. 

Now  the  pressure  during  the  recoil  is  limited 
to  a  given  maximum  consistent  with  the  packing  and 
therefore  the  effective  area  of  the  recoil  piston 
is  fixed.   With  large  guns  the  recuperator  reaction 
is  relatively  small  as  compared  with  the  maximum 
recoil  pressure,  and  therefore  the  intensity  of  the 
air  pressure  is  small.   Hence  the  recuperator  volume 
and  the  size  of  the  recuperator  is  large  as  compared 
with  a  separate  recuperator  system,  using  high  re- 
cuperator pressure  intensities. 

Thus  for  large  guns,  or  guns  with  low  elevation, 

separate  recuperator  systems  separate  from  the  brake 

system  usually  gives  a  smaller  recuperator  brake 
forging. 

Limitations  of  the  ratio  of  compression  "m". 

The  limitations  of   "m"  are  fairly 
fixed: 

(1)     The  minimum  "a"  is  based  on  & 
consideration  of  the  proper 
functioning  of  counter  recoil  at  all 


506 


elevations  . 

(2)     The  maximum  "m"  is  based  on  a 

consideration  of  horizontal  stability 
in  the  out  of  battery  position  for 
the  recoil,  as  well  as  heating  and 
rise  of  temperature  caused  by  the 
compression  of  the  air. 

(1)     With  guns  shooting  at  high  elevation,  the 
recoil  must  be  shortened  for  clearance  at  high 
elevations  and  lengthened  for  stability  at  hori- 
zontal elevation.  Thus  high  angle  guns  require  a 
variable  recoil,  the  ratio  of  short  to  long  recoil 
being  usually  from  one  half  to  two  thirds.   The  re- 
cuperator reaction  at  maximum  elevation  must  be  suf- 
ficient to  bring  the  gun  into  battery  with  a  moderate 
velocity  in  order  that  the  time  of  counter  recoil 
at  maximum  elevation  may  not  be  too  long.  This 
feature  is  of  considerable  importance.  Raising  the 
air  pressure  in  the  recuperator,  though  it  will 
sufficiently  accelerate  the  gun  at  maximum  elevation, 
will  give  too  great  a  velocity  at  horizontal  recoil 
and  thus  endanger  counter  recoil  stability.  Thus 
in  the  initial  design  it  is  important  that  the 

initial  volume  is  such  that  it  will  give  the  proper 

.         . 
ratio  of  compression. 

The  mean  recuperator  reaction,  or  rather  the 
recuperator  reaction  at  the  middle  of  the  recoil, 
was  shown  in  the  discussion  on  counter  recoil  to 

be, 

t 


*  v* 


s 

where  vh  •  the  max.  velocity  a+  horizontal  recoil 

(ft/sec) 

vg  -  the  max.  velocity  at  max.  elevation 
(ft/sec) 
total  recoil  packing  friction 


507 


B  »  coefficient  of  friction  from  0.1  to  0.2 
For  a  preliminary  design  constant,  we  nay  assume, 

vb  -  3.5  ft/sec.      va  *  2.5  ft/sec. 
and  taking  a  large  value  of  n  *  0.3  to  compensate 


%>-0.31!r+2Wr{sin01|-0.3(l+cos0m)]  (Ibs) 

With  a  hydro  pneumatic  recoil  system,  ire  have 

roughly,  2.5(PVB-Pyi)»P>vf-F'vi  hence  the  minimum  allowable 

ratio  of  compression,  becomes, 

*vf    2.5%.-1.5Fvi    1.5(1. 66Fva  -  Pyi) 

m   9   =   3   

?vi        pvi  Fvi 

of  course  the  ratio  nay  be  decreased  by  using  lower 
values  of  va  or  higher  values  of  v^,  or  both  but  the 
above  assumed  values  give  a  satisfactory  counter 
recoil  at  all  elevations. 

(2)     The  maximum  value  of  m  is  based  on  the 
following  considerations:- 

(a)  Horizontal  stability,  where 
a  high  final  air  pressure  nay 
exceed  the  allowable  overturn- 
ing fo^ee  consistent  with  stabil- 
ity in  the  out  of  battery 
position. 

(b)  The  maximum  allowable  c 'recoil 
buffer  pressure  which  linits 

the  potential  energy  stored  in 
the  recuperator  in  the  out  of 
battery  position. 

(c)  The  allowable  rise  of  tem- 
perature caused  by  the  com- 
pression of  the  air. 

With  light  mobile  field  carriages  ,  stability 
is  very  often  the  determining  factor  for  the 
maximum  allowable  ratio  of  compression.  This  is 
likely  to  especially  occur  when  the  mount 
elevates  to  very  high  angles  and  perfect  horizontal 


508 


stability  is  required  as  in  anti-aircraft  material. 
If  the  resistance  to  recoil  consistent  with  stabil- 
ity at  horizontal  recoil  is  small,  and  the  initial 
recuperator  reaction  large,  a  high  compression 
ratio  will  cause  the  total  resistance  to  recoil  in 
the  out  of  battery  position  at  horizontal  recoil 
to  be  greater  than  the  balancing  stabilizing  moment. 

Obviously  this  critical  condition  will  only 
occur  with  guns  of  high  elevation  and  required  to 
meet  rigid  horizontal  stability  limitations.   In 
an  ordinary  recoil  system  as  it  is  impossible  not 
to  have  more  or  less  throttling  at  the  end  of  re- 
coil, we  must  have  the  maximum  allowable  re- 
cuperator reaction  a  fraction  of  the  total  pull  for 
minimum  elevation  of  stability  0j. 

Therefore  the  maximum  allowable  ratio  of  com- 
pression from  a  stability  consideration,  becomes 

fvf   0.8[Kh+Wr(sin0i-0.3cos0i)] 
mmax  *  •=—  »  (min.  elev.) 

0.8(Rh-0.3Wr)        Kh 
»        "»     =  0.75  — —   (horizontal 

Fvi  Fvi   elevation) 

Therefore,  when  Fvi  is  large  and  Kh  small  as 
with  guns  for  high  elevation  and  rigid  stability 
requirements  "a"  becomes  small  and  low  ratio  of 
compressions  with  corresponding  larger  recuperators 
are  required.   Very  often  in  anti-aircraft  material 
"m"  becomes  smaller  than  that  required  for  proper 
counter  recoil  functioning  at  max.  elevation.   In 
such  a  case  it  is  preferable  to  sacrifice  horizontal 
stability  somewhat  and  increase  the  horizontal  re- 
sistance to  recoil. 

The  previous  formula  may  be  expressed  direct- 
ly in  terns  of  stability.   If 

Ws  *  weight  of  the  total  mount  (Ibs) 


509 


13  =  horizontal  distance  from  spade  to  Wg 

(ft) 
bh  =  length  of  recoil  at  horizontal  elev. 

(ft) 
£.£  *  min.  angle  of  elevation. 

We  have  for  the  max.  compression  ratio  based  on 
stability, 

0.8[(Wgls-Wr  bbcosefi) 

m  = 

M 

and  at  horizontal  recoil, 
0.8[Wsls-Wr(bh-0.3)) 


, 

0.75  ( 


rvi 

In  counter  recoil  systems  using  some  form  of 
a  c 'recoil  regulator  of  a  buffer  type,  we  bave  a 
necessary  geometrical  limitation  in  the  maximum  area 
of  the  buffer.   Thus  in  filling  in  types  of  buffers 
as  in  the  Schneider  and  Filloux  recoil  systems  as 
well  as  ordinary  spear  buffers  which  enter  the  pis- 
ton rod  at  the  end  of  c 'recoil,  the  effective 
area  must  necessarily  be  considerably  less  than  the 
area  of  the  piston  rod. 

With  spear  buffers  attached  to  the  piston  due 
to  void  considerations  at  the  beginning  of  the  re- 
recoil,  we  again  are  limited  in  a  large  effective 
buffer  area.   If 

Pb  max  *  the  max.  average  allowable  buffer 
pressure  (Ibs/sq.in) 

b  =  length  of  recoil  (ft) 

Ab  »  effective  area  of  buffer 

db  =  length  of  buffer  c 'recoil  (ft) 

Rp  '  total  packing  friction   (Ibs) 

WQ  *  total  potential  energy  of  the  recuperator 

(ft. Ibs) 

then,  when  the  counter  recoil  brake  comes  into 
action  towards  the  end  of  c 'recoil,  as  with  a  spear 
buffer,  we  have, 


510 


Pb  aax  *  " and  where  the  counter 

Abdb          recoil  brake  is  ef- 
fective throughout  the  counter  recoil, 

•0-(Wpsin0+Rp)  f 

Pb  max  "  '     ~^-       "h«re  V 

•  *  ratio  of  compression 

?o  •  initial  volume  of  recuperator  (cu.ft) 

Fvi  -  initial  recuperator  reaction  (Ibs) 

k  *  1.1  or  1.3  depending  whether  air  is  in  contact 

with  oil  or  separated  from  it  by  a  floating 

piston. 

The  expansions  for  pD  assume  a  constant  buffer 
fere*  during  the  buffer  action.   This  however  is 
not  always  the  case  and  therefore  the  above  ex- 
pressions should  be  multiplied  by  a  suitable 
constant  to  take  care  of  the  peak  in  the  buffer 
pressure  when  the  buffer  pressure  is  not  constant. 

It  is  to  be  particularly  noted  that  the  peak 
buffer  pressure  may  greatly  exceed  the  average 
buffer  pressure  as  obtained  by  the  above  expressions 

Combining  the  above  expressions,  we  have 


.ax 


,  „ 
This  is  a  very  important  limitation  for  m  and  is 

inherent  for  all  direct  acting  counter  recoil  buffer 
brakes.  Values  of  pb  max  range  as  high  as  8000  to 
10000  Ibs/sq.in.  with  short  spear  buffers  but  such 
pressures  should  not  be  tolerated  on  future  designs. 

In  general  the  c 'recoil  buffer  pressure 
should  be  maintained  as  low  as  possible,  thus  sim- 
plifying the  design  of  a  counter  recoil  system; 
therefore,  the  lower  value  of  m  consistent  with  a 
satisfactory  functioning  of  c 'recoil  at  all 
elevations  should  be  used. 
(3)     Though  the  total  energy  dissipated  in  a  re- 


511 


coil  cycia  Bust  necessarily  equal  the  initial  recoil 
energy,  it  is  important  to  distribute  the  energy 
in  the  parts  of  the  system  where  radiation  is  most 
effective,   if  the  energy  is  dissipated  entirely 
in  the  throttling  both  on  recoil  and  counter  recoil 
ire  have  a  large  nass  of  oil  with  corresponding  radi- 
ating surface.   With  high  compression  ratios  the 
air  in  the  recuperator  rises  to  a  high  temperature, 
which  nay  cause  injury  to  the  packing  and  lubrication, 
and  therefore  it  is  important  to  Maintain  a  low  com- 
pression ratio  and  thus  decrease  the  localized  heat- 
ing in  the  recuperator  where  radiation  is  the 
smallest  4 

As  to  the  allowable  rise  of  temperature  to  be 
permitted,  depends  greatly  upon  the  type  of  pack- 
ing to  be  used  and  the  packing  specification  should 
state  the  allowable  temperature  rise. 

Tne  temperature  T  at  the  end  of  a  recoil  stroke, 
above  the  mean  temperature  Tm  at  the  beginning  of 
the  stroke,  may  be  obtained,  from  the  relation, 


T.    Pai 

Assuming  a  ratio  2,  and  a  mean  temperature  25° 
centigrade,  we  have 

T  =»  298  x  2°*23  =  349°,  when  k  «  1.3  and  therefore 
the  rise  of  temperature  becomes,  T-Tm=51°C  or  92°F. 

The  temperature  rise  increases  considerably 
with  the  ratio  •,  thus  when  •  «  2.5,  T  -  TB  »  ?a°C 
or  ISff'F. 

RECDPERATOR  DIMENSIONS       With  hydro  pneumatic 
AND  LIMITATIONS.          recuperators  we  have 

two  or  more  cylinders, 

the  recuperator  cylinder  and  the  air  tank  or 
cylinder. 


512 


Let  b  =  length  of  recoil 

Ay  =  effective  area  of  recuperator  piston 
Aa  »  cross  section  area  of  air  cylinder 

Paf 
m  =  —  ratio  of  compression 

Pai 

Aa 
r  = — j[-  =  ratio  of  recuperator  cylinders. 

1  =  length  of  air  volume  in  terms  of  cross 
section  area  of  air  cylinder 

j  =  -  a  length  of  air  volume  in  terms  of  recoil 

b    stroke. 
Then,  the  initial  volume  becomes, 

nk  Aa      1 

V0  x  A_l  =  A_b  but  since  —  »  r:  -  s  j 

-  Av 

,k  -  l  V 


m 


I 

i 

rj 


hence        k 


i 

M*  -1 

When  a  floating  piston  separates  the  oil  and  air, 
k  =  1.3  (approx.)  Whan  the  oil  is  constant  with 
the  air,  k  *  1.1  (approx.) 

§  ^a  _  1   •* 

'  *    j   i 

•  k  -1 

1        i 

k 


•k  (r-1)  -  r 

i 

k    r 


513 


Tables  for  a  and  r  for  various  air  column  lengths 
when  k  =  1.3  are  given  below: 


r 

r 

r- 

•1.66 

-1.66 

log 

1.3   log 

• 

r 

3 

i. 

34 

2. 

239 

.  35005 

.  4550*7 

2. 

851 

3.5 

i. 

84 

1. 

902 

•  2*7921 

•36297 

2. 

307 

4. 

2. 

34 

1. 

209 

.  232*74 

•  30256 

2. 

007 

4.5 

2. 

84 

1. 

585 

•  2O003 

•  26004 

1. 

820 

5. 

3. 

34 

1. 

497 

.  1*7522 

.22779 

1. 

69O 

5.5 

3. 

84 

1. 

432 

•15594 

.20272 

1. 

595 

6. 

4. 

34 

1. 

382 

.14051 

.18266 

1. 

523 

r 

r 

r- 

1.25 

-1.25 

log 

1.3    log 

• 

r 

3 

i. 

"75 

i. 

"714 

.23401 

.30421 

2. 

015 

3.5 

2. 

25 

i. 

556 

.  142O1 

.24961 

1. 

777 

4.O 

2. 

•75 

1. 

455 

.16286 

.21172 

1. 

628 

4.5 

3. 

25 

i. 

385 

.  14145 

.18389 

1. 

527 

5. 

3. 

•75 

1. 

333 

.  12483 

.16228 

1. 

453 

5.5 

4. 

25 

1. 

294 

.11193 

.14851 

1. 

398 

c. 

4. 

75 

1. 

263 

.  10140 

.13182 

1. 

355 

514 


r 

r-1 

r 
(r-1) 

log^T 

1.31  log 

^1    " 

3.00 

2.00 

1.5000 

.  17609 

.  22892 

1.694 

3.50 

2.  50 

1.4000 

.  14613 

.18997 

1.549 

4.  00 

3.00 

1.3333 

.  12483 

.  16228 

1.453 

4.  25 

3.25 

1.3077 

.  11611 

.15094 

1.416 

4.50 

3.50 

1.  2857 

.  1O924 

.  14201 

1.387 

4.75 

3.75 

1.2667 

.  10278 

.13361 

1.360 

5.oo 

4.00 

1.  2500 

.09691 

.12598 

1.337 

5.50 

4.  gO 

1.  2222 

.os7o7 

.11319 

1.  298 

6.00 

5.00 

1.  2000 

.07918 

.10293 

1.267 

3 

2.  156 

1.391 

.14333 

.  18633 

1.536 

3.5 

2.  656 

1.318 

.  11992 

.15590 

1.432 

4.O 

3.156 

1.267 

.  10278 

.13361 

1.360 

4.5 

3.656 

1.231 

.09026 

.11734 

1.310 

5*0 

4.156 

1.20} 

.08027 

.10435 

1.271 

5.5 

4.  656 

1.181 

.07225 

.09393 

1.242 

6. 

5.156 

1.164 

.06595 

.08574 

1.218 

515 


516 


517 


r      r-.71! 

5      --.715 

log 

1.3   log 

i 

r 

3.         2.285 

1.313 

.  11826 

.15374 

1.425 

3.5    2.785 

1.257 

.09934 

.12914 

1.349 

4.0     3.285 

1.  218 

.09565 

.11135 

1.  292 

4.5    3.785 

1.  189 

.07518 

.09773 

1.  252 

5.0     4.  285 

1.167 

,o67o7 

.08719 

1.222 

5.5    4.785 

1.  149 

.06032 

.07842 

1.198 

6.       5.285 

1.135 

.05500 

.07150 

1.179 

Vo~  lAvb 

b 

'.-  7  M 

pvdV  =  |nr 

•Vjj+WpCsin^ 

nax+u   cos2fBax 

1U4A                                        HI  a  A 

'!-e 

where   VB  * 

2   ft/sec, 

i 

roughly. 

Now  pvVk»pvi 

v|; 

fcence   py   =  pvi  Vk — k 
hence 


PV1   ^      V0-.375A¥b     7k 


where   k  =   1.1   to 
1.3 


-    (V0-.375    A. 


B 


1-k 

The  solution  of  this  expression  is  com- 
plicated and  trial  values  of  VQ  nay  be  substituted 
more  easily.  _  , 


Knowing  VQ  and  Vf  =  VQ-Ayb,  we  have  • 


V 


Values  of  m  greater  than  this  value  are  en- 
tirely unnecessary  for  satisfactory  functioning  »t 
counter  recoil  at  all  elevations.   When  the  initial 
value  of  the  recuperator  reaction  is  »ade  greater 


518 


than  thai  required  to  bold  tba  gun  in  battery,  the 
necessary  ratio  of  m  decreases  in  tbe  limit  if  m  » 
1,  then 

* 


Kvi*PviAv 


4mrv 


Due  to  the  uncertainty  and  variation  of  both 
packing  and  guide  friction,  an  excess  initial  re- 
cuperator reaction  is  always  used  and  thus  even  for 
very  low  values  of  "•"  we  usually  have  in  modern 
artillery  a  surplus  of  potential  energy  in  tbe  re- 
cuperator. 

glilRAL  DlSiaH  LIMITATIONS. 


SURVEY  OF  LIMITATIONS     The  design  limitations 
IN  CARRIAGE  DESIGN.    for  a  gun  mount  depend 

primarily  of  course  on  the 
?  .  particular  use  to  be  obtained 

from  the  gun  and  the  general 

type  of  carriage  to  be  used.   Though  each  design 
la  a  problem  by  itself,  it  is  however  possible  to 
derive  and  point  out  certain  broad  limitations 
that  lust  be  observed  for  a  satisfactory  design. 

The  fundamental  requirements  and  limitations 
for  the  various  classes  of  mounts  are  considerably 
different.  The  question  of  elevating,  traversing, 
etc.  certain  more  strictly  to  a  given  mount.   Row- 
tVer,  certain  broad  limitations  apply  to  tbe  various 

olaases  of  mounts  and  for  good  design  these  limit- 
ations must  be  always  considered  quite  independent 
of  tbe  requirements  for  the  particular  service  of 
the  gun. 

(1)  For  mobile  mounts  minimum  weight 
and  stability  under  firing  conditions 
are  primary  limitations. 

(2)  For  caterpillar  mounts  minimum 
weight  and  stability  under  firing 


519 


conditions  are  again  primary 
limitations. 

(3)  For  railway  mounts,  due  to  size 
and  cost  of  parts,  minimum  weight 
consistent  with  stability  is  ia- 
portant  but  other  factors  such  as 
clearance,  method  of  loading,  etc. 
have  perhaps  more  influence  on  the 
design. 

(4)  For  stationary  mounts  for  defense 
work  stability  is  easily  secured  and 
though  it  is  highly  desirable  to  keep 
the  size  and  weight  of  parts  as  small 
as  possible,  the  vital  factors  are 
accessibility,  ease  in  loading  and 
endurance. 

LENGTH  OF  RECOIL        The  strength  of  a  gun  car- 
AT  HA XI MUM  ELEVATION  riage  depends  roughly  on  the 
AND  MAXIMUM  RECOIL    maximum  recoil  reaction. 
REACTION.  How  the  recoil  reaction  varies 

roughly  inversely  as  the 

length  of  the  recoil  for  a  given  recoiling  mass  and 
ballistics;  therefore  it  is  highly  desirable,  for 
lower  stresses  in  the  carriage,  to  maintain  as  long 
a  recoil  as  possible.   But  at  maximum  elevation 
we  are  immediately  limited  by  clearance  *of  the  gun 
striking  the  ground  or  platform.   As  the  height  of 
the  trunnions  and  axis  of  the  bore  are  fixed  by 
stability  at  horizontal  elevation  clearance  in 
traveling  and  accessibility  for  loading,  the  recoil 
at  maximum  elevation  (as  well  as  the  maximum  recoil 
reaction)  becomes  definitely  limited. 

Means  for  increasing  the  recoil  and  thereby 
diminishing  the  recoil  reaction  are  as  fellows: 

(1)     By  digging  a  pit  under  the  gun. 
(2)     By  placing  the  trunnions  as  far 


520 


as  possible  to  the  rear  adjacent  to 
the  breech  end  of  the  gun  and  balanc- 
ing the  tipping  parts  by  the  use  of 
a  balancing  gear. 

(3)     By  raising  the  trunnions  as  the 
gun  elevates,  obtaining  a  low  height 
of  the  trunnions  above  the  ground 
when  stability  is  required  and  a 
high  position  when  stability  is  no 
longer  a  requirement  and  a  long  re- 
coil is  desired. 

LENGTH  OP  RECOIL  AT       As  mentioned  before, 
MINIMUM  ELEVATION     howitzers  are  designed  for 
STABILITY.  high  angle  fire,  ranging 

roughly  from  20  to  70  de- 
grees. Therefore,  stability 

is  not  of  great  importance  up  to  20  degrees  ele- 
vation.  At  this  elevation  the  moment  arm  of  the 
overturning  force  ,    about  the  trail  support  be- 
comes small,  and  therefore  it  is  possible  to  con- 
siderably raise  the  trunnion  and  thereby  lengthen 
the  recoil  at  maximum  elevation  than  with  guns. 
Further  for  a  given  height  of  trunnions"  the  length 
of  recoil  can  be  shortened  for  an  elevation  of  20° 
consistent  with  stability.   Thus  with  howitzers, 
it  is  possible  to  maintain  a  constant  recoil  length 
for  all  elevations.   This  is  of  more  or  less  ad- 
vantage in  simplifying  the  recoil  system. 

With  a  gun,  the  elevation  ranges  roughly  from 
0°  to  50°.   At  0°  elevation  the  overturning  moment 
about  the  spade  support  is  a  maximum,  and  the 
stabilizing  moment  a  minimum.   (See  Chapter  III). 
Hence  a  long  recoil  is,  essential  in  order  to  reduce 
the  recoil  reaction  and  overturning  moment. 

The  maximum  horizontal  recoil  however  is 
limited,  due  to  the  fact  that  at  the  end  of  recoil 


521 


though  the  overturning  moment  is  decreased  by 
lengthening  the  recoil,  the  stability  moment  is 
also  decreased  in  the  out  of  battery  position  due 
to  the  recoiling  mass  being  displaced  to  the  rear. 
Thus  we  arrive  at  an  initial  length  of  recoil 
where  further  increase  causes  a  decreased  stability. 
If   Hs  =  weight  of  carriage  and  mount  together 

(Ibs) 

Rh  =  horizontal  recoil  reaction   (Ibs) 
Vf  *  max.  velocity  of  free  recoil  (ft/sec) 
wr  =  weight  of  recoiling  parts   (Ibs) 
b  »  height  of  axis  of  bore  above  ground  (ft) 

Then          w 

0.47  "r  T» 
Rb  *  — ~  T~  M   (approx.). 

then  Rh  h  +  Wr  b  =  Hgls  at  critical  stability.   Now 
the  actual  overturning  moment,  becomes, 

.  ,fr  b   and  the  corresponding  stability 
b  g 

moment  =  "s^s 

If  we  differentiate  the  actual  overturning 
moment  with  respect  to  b  and  equate  to  zero,  we  ob- 
tain, the  maximum  allowable  horizontal  recoil  for 
a  given  recoiling  weight,  hence 

0.47W.VJ  h            0.47W.VJ  h 
d( £-1—  +  w  b)  , L£_  +  Wp  »  0 

d  b  b*g 

hence  bb  max  -  0.121  Vf  /~h~ 

Another  limitation  on  the  length  of  recoil 
at  horizontal  elevation,  is  due  to  the  fact  that 
as  the  recoil  lengthens,  the  distance  between 
the  clip  reactions  decreases,  and  the  clip  re- 
actions and  the  guide  frictions  become  excessive 
in  the  out  of  battery  position  due  to  the  over- 
hanging weight  of  the  recoiling  parts.   Such  ex- 


0.471»r  V  h 


522 


cessive  guide  friction  caused  by  the  moment  of  the 
overhanging  weight  combined  with  the  recuperator 
reaction  at  the  beginning  of  counter  recoil  may 
prevent  satisfactory  return  into  battery, 

Further  the  bending  moment  at  the  rear  clip 
reaction  of  the  gun  becomes  excessive  due  to  the 
large  overhang  in  addition  to  the  recoil  pull  on 
the  gun  lug.   Thus  the  length  of  horizontal  recoil 
is  limited  by  the  minimum  allowable  distance  between 
clip  reactions  when  the  gun  is  out  of  battery.   If, 
with  this  maximum  recoil  the  mount  is  unstable, 
either  the  weight  of  the  mount  oust  be  increased 
or  outriggers  reaching  further  out  must  be  used. 
But  for  mobile  mounts  minimum  weight  is  essential, 
hence  extended  outriggers  or  increase  of  trail 
length  must  be  resorted  to.    As  the  gun  elevates, 
stability  increases  and  the  recoil  may  be  shortened 
consistent  with  clearance  and  stability. 

Kith  anti-aircraft  guns,  it  is  desirable  to 
shoot  from  0°  to  80°  since  the  piece  must  be  inter- 
changeable for  field  work  if  necessary.  Therefore 
the  limitations  on  anti-aircraft  material  are  more 
pronounced  and  the  change  of  length  of  recoil  is 
greater  from  0°  to  max.  elevation  than  with  other 
types  of  mounts. 

RECOILING  WEIGHT  FOR     The  weight  of  carriage 
MINIMUM  WEIGHT  OF  GUN   proper  not  including  the 
CARRIAGE.  recoiling  mass  is  more  or 

less  proportional  to  the 
necessary  strength  re- 
quired in  the  carriage.    Now  the  strength  of  the 
carriage  is  roughly  proportional  to  the  maximum 
recoil  reaction.   Further  the  weight  of  a  car- 
riage depends  upon  the  type  or  configuration  of 
the  mount.    Hence,  for  any  given  type  of  car- 
riage the  weight  is  roughly  proportional  to  the 
maximum  recoil  reaction.   If,  therefore,  a  given 


523 


type  of  carriage  is  designed  to  withstand  a  given 
recoil  reaction,  the  higher  the  carriage  is  stressed, 
the  smaller  becomes  the  ratio  of  the  weight  of  the 
carriage  to  the  recoil  reaction.   Therefore  the 
weight  of  efficiency  for  a  particular  type  of  mount 
is  increased  by  decreasing  the  weight  of  the  mount 
per  given  length  of  recoil.  Obviously  if  a  given 
type  of  mount  was  designed  so  that  all  its  parts 
were  stressed  to  the  elastic  limit  for  the 
maximum  recoil  reaction  we  would  have  the  minimum 
possible  weight  for  the  given  type  of  carriage. 
Let  wc  -  weight  of  the  carriage  mount  not  including 

the  recoiling  mass. 
R  *  the  maximum  recoil  reaction. 
c=  weight  of  the  carriage  mount  proper  when 

stressed  to  the  elastic  limit, 
k  =  the  weight  constant  for  the  carriage  mount 

proper. 

k1  *  the  weight  constant  when  stressed  to  the 
elastic  limit. 

Then  ' 

c    >  i  '    c 

k  •  r  and  k  '  r 

Obviously  the  weight  efficiency  in  a  given  design 
pertaining  to  a  given  type  of  mount,  becomes, 

k    "c 
weight  off.  *  —  3  — 

k'   Wg 

Now  the  weight  efficiency  varies  considerably 
with  the  type  of  carriage  used,  certain  types  hav- 
ing considerably  more  dead  weight  than  other  types. 
Further  the  weight  efficiency  depends  directly  on 
the  factor  of  safety  recommended  in  the  design. 

A  table  for  the  constant  "k"  for  various  types 
of  mounts  is  given  below: 


524 


Weight 

Re. 

Max. 

Length 

Weight 

Weight 

of 

coil- 

Recoil 

of  Re- 

of 

Con- 

Sy«- 

ing 

React- 

coil. 

Mount 

stant 

te«. 

wt  . 

ion. 

not  in- 

of 

clud- 

Car- 

ing 

riage. 

Recoil- 

ing Wt. 

Carriage 

W. 

»r 

R 

L 

We 

K 

3'Model  of 

2520 

960 

4923 

45 

1560 

.317 

1902. 

75  -/» 

265*7 

1050 

5250 

49 

1607 

.306 

Tr  e  no  h 

M.  189*7. 

75«/»  v.of 

3045 

911 

12100 

46 

2134 

.176 

1916. 

and 

18 

3.3"  g«n 

43*72 

1435 

2100 

45 

2937 

.146 

Carriage. 

and 

30 

3  .  8  "  How.  C  «r- 

2040 

935 

13750 

4O 

1105 

.08 

r  iage,  1915. 

a  ad 

22 

4.7"  Sun, 

7420 

2745 

17500 

70 

5675 

.324 

v.  1906. 

4.  7'How.O»r- 

3988 

1372 

19430 

52 

2616 

.135 

r  iage,  1908. 

and 

24 

155«/m  He*. 

7600 

3498 

390OO 

51.4 

4100 

.  105 

8  o  ha. 

155-/-1'  il- 

19860 

9050 

66000 

43 

10810 

.164 

leaz  . 

and 

71 

8"  Vioker«, 

2OO48 

9356 

11730O 

52 

1O692 

.091 

Mk.  Til. 

and 

24 

24O  •/• 

41296 

15790 

15OOOO 

46. 

25526 

.171 

8CHIIIDBR. 

To  give  a  farther  physical  conception  of  the 
meaning  of  "k"  we  note  from  previous  calculations 
that  the  155  m/m  Filloux  is  extra  strong,  most  of 
the  fibre  stresses  not  exceeding  10,000  Ibs.  per 
sq.  in.   Comparing  it  with  the  3.3  inch,  a  somewhat 
similar  type  of  mount  we  would  expect  the  3.3  inch 
to  be  well  stressed.   This  is  actually  the  case. 
Two  very  similar  types  of  heavy  field  trail  car- 
riages are  the  8"  dickers  and  155  m/n  Schneider, 
both  having  the  same  type  of  trail.   Both  car- 
riages are  well  designed,  having  in  the  various  parts 
about  the  same  maximum  fibre  stress.  Therefore  as 
we  would  expect  the  constant  "k*  is  approximately 
the  same.  The  3"  Model  1902  is  not  efficiently  de- 
signed as  compared  with  similar  types  such  as  the 
75  m/m  M.1916.   We  thus  see  that  "k"  when  compared 
with  types  of  similar  carriages  gives  us  a  crude 
idea  as  to  the  efficiency  of  the  design  of  the 
carriage  itself. 

Now  the  weight  of  the  system  is  the  recoiling 
weight  plus  the  weight  of  the  mount  proper  (i.  e. 
the  stationary  parts),  that  is  wg  »  wr  +  wc 

where  ws  =  the  weight  of  system 
wr  =*  recoiling  weight 
we  =  weight  of  stationary  parts,  or  mount 

proper. 

Per  a  given  type  of  mount,  the  weight  of 
carriage  may  be  assumed  roughly  proportional  to  the 
recoil  reaction,  that  is,  wc  =  k  R 

Now  from  the  principle  of  linear  momentum, 
neglecting  the  small  effect  of  the  recoil  reaction 
during  the  powder  pressure  period,  and  the  air 
resistance,  then  m  v  +  I  4700  »  m  ?  wnere  m  and  v  - 
mass  and  muzzle  vel.  of  projectile. 

I  4700  =  the  momentum  effect  of  the  powder 
gases,  hence        +  - 
V  = 


526 


but  R  *  — —     approximately. 
26 


(•v+I  4700)* 


2»rb 


k  u1        <<•**•  4700)' 

hence  R  »    —       where   k     *  - 


therefore   wft   s    k  R 


2b 

k   k  ' 


Now  for  minimum  weight  of  the  total  system, 
recoiling  parts  together  with  carriage  mount, 

dw  d(wr+-jU^) 

s  "r         kk 

r  —  »  0   that  ia  -  -  1  -  •—  »  0 

d"r  i«r  »*. 


w*-kk' 


or  w_  = 


•r 

"ft 

where   It  *  s"   obtained    from   table 


Jot) 


ballistic  constant 


To  use  the  above  fornula  ia  a  new  design  we 
take  the  value  of  k  fren  a  siailar  well  designed 
type  of  carriage,  using  a  somewhat  lower  value  of 
"k"  according  to  the  judgment  of  a  designer  IB 
improving  the  weight  efficiency  of  the  mount  proper 
over  a  similar  previous  design.   Knowing  the 

ballistics  of  the  new  mount,  we  find  a  very  definite 

weight  for  the  recoiling  mass. 

It  is  interesting  to  note  that  usually  the 

strength  curve  of  a  gun  say  be  considerably  increased 

if  the  proper  weight  of  recoiling  mass  consistent 

with  minimum  weight  is  used. 


C  H  A  P  T  B  B   VIII. 


This  chapter  contains  a  discussion  of 
some  of  the  types  of  hydro-pneumatic  recoil 
systems  with  calculations  of  characteristics  of 
service  designs. 

It  has  been  found  desirable  to  print 
this  chapter  separately. 


527 


CHAPTER   IX. 

HYDRO-PNEUMATIC  RECOIL  SYSTEMS. 
(Continued) 

SCHNEIDER  RECOIL         The  Schneider  recoii 
SYSTEM.  system  consists  of  an  in- 

dependent recuperator  sys- 
tem of  a  hydro-pneumatic 
type.    The  cylinders  are 

in  one  forging  and  are  secured  to  the  gun.   The 
cylinder  forging  is  known  as  the  sleigh  or  slide 
and  recoils  with  the  gun.   The  brake  and  recuperat- 
or rods  are  held  stationary  and  attached  at  their 
ends  to  a  yoke  on  the  cradle.   The  hydraulic 
brake  piston  rod  is  hollow  and  contains  a  filling 
in  buffer  chamber.   Attached  to  the  sleigh  and 
sliding  within  this  buffer  chamber  is  a  counter 
recoil  buffer  rod.   The  throttling  during  the  re- 
coil is  effected  through  an  orifice  formed  by  the 
difference  in  areas  of  a  circular  hole  in  the  pis- 
ton and  the  area  of  the  buffer  rod.   For  varying 
the  throttling,  the  areas  of  the  buffer  rod  are 
tapered,  i.  e.  the  diameter  of  the  buffer  rod 
varies  along  the  recoil. 

The  recuperator  cylinder  consists  merely  of 
the  stationary  recuperator  piston  which  moves 
relative  to  the  forging  on  recoil.   The  recuperat- 
or cylinder  communicates  by  a  large  passage  way  to 
the  air  cylinder  partly  filled  with  air.   The  air 
cylinder  is  placed  forward  and  is  made  shorter 
than  the  recuperator  and  brake  cylinder.   This  is 
necessary  in  order  that  at  maximum  elevation  the 
oil  in  the  air  cylinder  covers  the  passage  way 
communicating  with  the  recuperator  and  air 
cylinders.    It  is  very  important  in  the  initial 

529 


530 


• 


531 


lay  out  of  the  Schneider  recuperator  system  that 
at  a  maximum  elevation  the  oil  completely  covers 
the  communicating  passage  way  in  the  air  cylinder 
and  the  recuperator  initial  volume  should  be 
reckoned  in  the  air  tank  beyond  this  oil  cover- 
ing.  The  passage  way  is  made  sufficiently  large 
so  that  we  have  practically  no  throttling  in  the 
recuperator  system. 

During  the  recoil,  figure  (  I  ),  the  brake 
throttling  is  effected  primarily  through  an 
orifice  formed  by  the  counter  recoil  rod  in  a 
circular  hole  in  the  piston.   The  simultaneous 
compression  of  the  air  recuperator  during  the  re- 
coil takes  place  practically  along  an  isothermal 
curve,  due  to  the  fact  that  oil  and  air  are  in 
direct  contact  in  the  recuperator.   It  has  been 
found  by  careful  computation,  however,  that  an  ex- 
ponent equal  to  1.1  gives  a  close  approximation 
in  the  compression  curve  of  the  air  and  the  com- 
pression of  recoil  in  the  brake  cylinder.   The 
buffer  is  filled  by  the  pressure  head  in  the  re- 
coil cylinder,   the  oil  passing  through  fairly 
large  orifices  in  the  buffer  head  FF,  the  slide 
of  the  buffer  head  being  away  from  the  counter 
recoil  buffer  rod,  see  figure  (  I  ). 

During  the  counter  recoil  the  slide  on  the 
buffer  bead  is  pushed  in  contact  with  the  buffer 
rod,  and  the  apertures  which  filled  the  buffer 

chamber  during  the  recoil  are  thereby  closed  and 
the  throttling  now  takes  place  through  new  orifices 
of  a  very  small  magnitude.    The  buffer  chamber 
having  been  completely  filled  during  the  recoil 
enables  us  to  have  a  continuous  regulation  through- 
out counter  recoil.   The  counter  recoil  throttling 
is  effected  through  a  constant  orifice  for  over 
half  of  the  counter  recoil.   We  then  have  a  taper- 
ing orifice  until  the  gun  nearly  reaches  the  in 
battery  position. 


532 


In  the  Schneider  system  the  recoil  is  designed 
constant  at  all  elevations  or  practically  so,  a 
slight  variation  taking  place  with  the  elevation. 
The  recoil  system  is  made  to  vary  according  to  the 
stability  slope  at  the  minimum  firing  angle  of  ele- 
vation. 

The  primary  advantages  of  the  Schneider  sys- 
tem are: 

(1)  An  increased  recoiling  mass  due 
to  the  recuperator  sleigh  contain- 
ing the  cylinders,  recoiling  with 
the  gun  and  thereby  decreasing  the 
reaction  on  the  carriage. 

(2)  The  simplicity  of  the  recoil 
mechanism,  especially  from  a 
fabrication  point  of  view. 

The  disadvantages  of  the  Schneider  system,  are: 

(1)  due  to  the  fact  that  the  primary 
element  of  simplicity,  the  throttling 
effected  through  a  simple  tapering 
counter  recoil  rod,  inherently  pre- 
vents any  possibility  of  a  variable 

recoil. 

• 

(2)  the  massive  sleigh  or  slide  at- 
tached to  the  gun,  though  reducing 
the  reaction  on  the  carriage,  lowers 
the  center  of  gravity  of  the  recoil- 
ing parts  below  the  axis  of  the  bore 
so  that  on  firing  a  large  load  is 
thrown  on  the  elevating  arc.   To  off- 
set this,   on  snail  caliber  guns  a 
counter  weight  has  been  mounted  on 
top  of  the  guns.   On  the  larger 
caliber  guns  as  in  the  240  m/m 
howitzer,  a  brake  clutch  was  intro- 
duced on  the  shaft  of  the  elevating 
pinion  which  slipped  during  firing. 


533 


534 


Further  tba  air  cylinder,  "being  necessarily 
placed  forward  of  tbe  recuperator  brake  cylinders 
with  a  long  recoil  gun  requires  a  very  long  forg- 
ing and  corresponding  guides  on  the  cradle. 

On  tbe  wbole  the  Schneider  recoil  system  has 
proved  one  of  the  most  satisfactory  recoil  systems 
used  during  the  late  war,  being  simple  to  fabricate 
and  tborougbly  rugged,  due  to  its  simplicity  in  de- 
sign. 

Example  and  calculation  of  tbe  Schneider  recoil 
system  for  the  240  m/m  Howitzer:   As  an  example  of 
a  satisfactory  recuperator  brake  especially  adaptable 
for  a  howitzer,  calculations  in  tbe  design  layout 
of  tbe  240  m/m  howitzer  recoil  system  are  given  in 
tbe  following:- 

BIOOIL  CALCULATIONS  240  M/M  SCHMKIDBR  HOWITZER. 

Type  of  gun  -  240  m/m  howitzer 

Total  weigbt  at  recoiling  mass  *  15,790  Ibs.  *  Kr 

Muzzle  velocity  -  1700  ft/sec.  *  Vm 

Length  of  recoil     B"     44,833     46,73 

Angle  of  elevation   0        10°       60° 

-i                              pa 
Intensity  initial  air  pressure  Pal  »  -— *  576 

•7854  "a    ik./ 
Initial  air  pressure  -  .7854PaiD*  »18800  Ibs.    sq.in, 

Height  of  axis  of  bore  from  ground— 43" 

w  V* 

Mean  constant  pressure  Pa  *  »  1,189  x  10  Ibs. 

64.40 

Weight  of  powder  charge  if 40  lb«. 

Travel  of  projectile  in  bore  -  u  -  160" 

Maximum  powder  pressure  on  base  of  projectile  Pm  = 
2005  x  10'  Ibs. 


535 


Maximum  pressure  on  breech  Pb  »  1.12  Pm 

Initial  air  volume  V°  =  2970  cu.in. 

Final  air  volume  Vf  =  V°  cu.  in  -  Ae  b"  =  1510 

Vi  t 

Final  or  maximum  air  pressure  »  p.*  *  pa4 ( — ) 

Vf 


INTERIOR  BALLISTICS. 


e  ='  twice  abscissa  at  maximum  pressure 

D 


-  i  ± 
16  Pe  16  Pe 


muzzle  pressure  on  base  of  breech 

£Z.e*  —^ m  pfe  622,000 

4     (e+u)* 


Velocity  of  free  recoil 
w  V   +  4700  if 


50.25ft.sec. 


Velocity  of  free  recoil  -  projectile  leaving  muzzle 

w  Vm  +  .5w  V_ 
yo  ,  » 2.  40.15ft.sec. 


Time  of  projectile  to  muzzle 

t    ,  i-JiL 

*        *    12   Vm 


.01175  sec. 


536 


Time  of  expansion  at  free  gases 

2(Vf-V0  )    Wr 

t  »  «  .01538  sec 

P0b        32.2 


Free  movement  of  gun  while  shot  travels  to 
muzzle 


*   12(Wr+w+i) 


Free  movement  of  gun  during  powder  expansion 

pob  «<•* 
Xa  »  —  — -  +  V0t8  .7179ft. 

Wr    3 


Total  free  movement  of  gun  during  powder 
pressure  period 

Z  =  Xt  +  X,  1.0279 


Time  of  pressure  period 


T  »  tt  +  t^  .02713  sec. 


Total  resistance  to  recoil  in  battery 

mrvj  +  m(b-E)a 
K  «         '  •     "  '  Variable  recoil 

m  T* 

2[b-E  +VfT-  -  —  (b-E)] 
2  mr 


where  K  *  total  resistance  to  recoil  during  powder 
period   (Ibs) 

b  *  length  of  recoil   (ft) 
E  *  free  displacement  of  recoil  during  pow- 
der period   (ft) 


537 


T  =    total   powder   period      (sec) 

c  »r 

m  =     cos  0  stability  slope 

d 

c  =  constant  of  stability 

d  =  distance  from  line  through  center  of 
gravity  of  recoiling  parts  parallel  to 
bore  to  center  of  pressure  exerted  on 
spades. 


,  490     -  f 

g          32.2 

C  \fr   cos   0       cwr  0.85   x   15780 

m  =  -  -  -  =  —  (approx.  )  =  -  —  -  - 

d                       h  <j  .  oo 

«    >::*-f.-iM»iri-!^    f  **.-:•  "rV-  -*• 
3760 

E  =    1.0279  ft. 
T  -    .02713   sec. 

b   =   44'833       »   3.736  ft. 
12 

Hence 

490   x  50755*    +   3760(2.708)* 


K 


3760    .02713* 

2 (2. 708+50. 25x. 02713  x  x  2.708) 

2        490 

1264660 


8.13 


155000  Ibs. (approx) 


Total  resistance  to  recoil  out  of  battery 

Rt8 
k  -  K  -  m(b-  E  +  — ) 

2mr 

»  155000  -  3760(3.736-1.028+  15500°  x  -0271  ) 

2  x  490 


538 


»  155000  -  3760  x  2.824  »  144,000  Ibs. 


CALCULATION  OF  THE  VARIATIOH  OF  TH1  HI- 
ACTIOB  AIB  PRESSURE  IK  TH1  HBCOIL. 

Initial  air  volume  -  2970  cu.in. 

Initial  air  pressure  =  576  Ibs/sq.in) 

Length  of  recoil  (10°  elevation)  =  44.8  inches. 

Length  of  recoil  (60°elevation)»46.73 

Effective  area  of  recuperator  piston  =  35.766  Ibs. 

Effective  area  of  hydraulic  piston  =  31.2 

Final  Pressure       (Initial  volume) 

___^  .^.  _  —  —  —  —  »___^_——, 

Initial  pressure      (Final  volume) 

Final  volume  »  initial  volume  *  area  at  recuperat- 
or piston  x  length  of  recoil. 


.-.  Final  pressure  (10°  elevation)-  576(  -    -  )*•* 

2970-35.766x44.8 


'   =  576  x  2.345  -  1350  Ibs/sq.in 
1368 


Final  pressure  (60°  elevation) 
2  970 


576( 


2970  -  35.766  x   46.73 


,,2970.1.1 

-   576  x   2.49  -   1434   Ibs/sq.in. 
1.299 


For  40"  Recoil 

2970        *  "  * 

Final  pressure  *  576( ) 

2970-35.766  x  40 

576  x  2.065  *  1189  Ibs/sq.in. 


539 


1389  x  35.766  »  42525  (Plot  these  values  above  fric- 
tion) 


For  35"  Recoil 


Final  pressure  *  576  (-  ) 

2970  -  35.766  x  35 

576   *   1.815  =    1045  Ibs/sq.in. 
1045   «  35.766  «  37375 


For  30"  Recoil  2g?Q 

Final  pressure  »  576  ( ) 

2970  -  35.766  *  30 

576  x   1.643   »   946  Ibs/sqiln. 
946  x  35.766  -   33835 


For  25"  Recoil  2970        *•* 

Final  pressure  »  576  (  ) 

2970  -  35.766  x  25 

576  x  1.483  -  854  Ibs/sq.in. 
854  x  35.766  =  30544 

For  200  Recoll  ^^ 

Final  pressure  *  576  ( ) 

2970  -  35.766  x  20 

576  x   1.35  »   788   Ibs/sq.in. 
778   x  35.766  »   27825 


For  15-  Recoil          ^ 

Final  pressure  »  576( ) 

2970  -  35.766  x  15 


576   x   1.22  *    702    Ibs/sq.in. 
702    x  35.766  «   25107 


For  10"  Recoil 

Final  pressure  «  576(  -  ) 

2970  -  35.766  x  10 

576  x  1.155  =  665  Ibs/sq.in. 
665  x  35.766  «  23784 


540 


For  5"  Recoil  2970         *•* 

Final  pressure  «  576  ( ) 

2970  -  35.766  *  5 

576  *  1.072  »  617  Ibs/sq.in. 
617  x  35.766  -  22067 


Calculation  of  Velocity  Curve 


(During  Powder  Pressure  Period) 
Point  #1.  Coordinates  Vo  and  Xo 


Vo  '  V£o 


Kt 


o 


m 


r 


Kt 

l«o's; 


When  the  projectile  leaves  the 

muzzle, 

K  =  155000  total  resistance  to 


s 

recoil 

r 


u  *  travel  of  projectile  in  bore 

160" 
vo=  muzzle  velocity  =  1700ftsec. 

w  »  weight  of  shell  =  353 

w  =  weight  of  powder  charge  *  40 
Wr  *  weight  of  recoiling  parts 

15790 

m  -  -  =  490 
32 

(w  *  .5w)V0  3  u 

—  -  -   ;    t0-^- 

(353  +  .5  «  40  x  1700) 

=40-15 


3<16°     :  .01176 


2  x  12  x  1700 

...  ,0.-40.1S-155000>  -01175.  40.15-3.70! 

490 


36.449ftsec. 


541 


w    +   .5if  (353   +   .5   *   40)160 

Xfo  *  —  w  -  u  s   - 


15790   *   12 


155000  x    .01175 
"2   x   490~ 

.32  -   .0221  =    .2979   ft.=  3.57  inches. 


Point   #2. 

Maximum  restrained  recoil  velocity  and  correspond- 
ing orifice. 


T   * 


K(T-tQ) 


tm  »    .02713  - 


.01175) 


155000 (.02713- 
622000 


tm  »    .02329 


Vfn     »   40.15    *  6220°°    [.02329  -    .01175]!- 
490 


62000(. 02524  -    .    01175) 
4x490(50.25-40.15) 


40.15+9.328 


v       =   49.478 
f  m 


s  49.478  .  15500X.02329 
m  490 


.see. 


Xfm  "  2m. 


542 


fo  +  —  (*•-*(>>  -        ,  > 
mr  6mr(Vf-V0) 

Xf0  *  —  :  —  x  u  »  .32  (see  Point  II) 


.32+[40.15+6220°°  (.02329-.  01175)- 
490 


(.02329  -  .01175)' 


6x490(50.25-40.15) 

=  .32*. 632  =  .952  Ft*  11.42  in. 

155000  *  .02329* 

X.  »  .952 —  =  .952  -  .0855  » 

2  x  490 

.866ft. -10. 39  in. 
Point  *3  (At  end  of  the  powder  period) 


155000  »  .02713 


490 


Vr  »  4l.7ft.sec. 


X 

r  155000  x  .  02713* 

Xr  » 


2  x  490 
1.0279  -  .1155  -  .9124ft«  10.94in. 


Velocity  Curve  (during  retardation  period) 

/2CK-  •"-  (b+X-2Xr)0>-x)] 
Vx  «  /  

mr 

For  x  »  1.5  feet. 


543 

3760 
[155000 — <3. 89+1. 0279-2x. 9124)] (3.89-1.5) 

• 


490 


37.4  ft. per  sec. 
For  x  »  2  feet 


3760 
21155000  (3. 89+1. 0279-2 x. 9124)] (3.89-2) 

P. 

490 

33.6 


For  x  «  3  feet 


[155QOO ^(3. 89+1. 0279-2x. 9124)3  (3. 89-3) 

vx 

22.8 
For  x  -  3.73(total  recoil) 

Velocity  »  0 

_^_^^^___^____ 


Calculation  of  Guide  and  Packing  Frictions. 
g 


2oKdb 
Guide  friction  Rg  -  -  -  -    approx. 


u  -  .15 

K  =  155000 

dfc  *  15  .5  '(in)  distance  from  center 

of  gravity  to  resultant  pull. 
1  =  37  +  48=85"(in)  mean  distance 

between  clip  reaction. 

2*.  15x154725x15.  5 

.-.  Guide  friction  -  —  -  —  —  =  8450  Ibs 

85 


Stuffing  Box  Friction 


Recuperator  stuffing  box 
Diani.  *  2.169 


Bear  sleeve  -  .5"+. 875" 

contact 

Inner  packing  ring  -  .787 
Gland  -  .87 


544 


Recoil  stuffing  box 
diam.  =  4.728 


Rear  sleeve  -  .75*. 5 
>   Inner  packing  ring  .787 
Inner  gland  -      .866 


(Spring  pressure  +  0.1  pressure)(.75  diam.  x  H 
.09  x  length  of  contact)  Formula. 


1058 
Spring  pressure  from  drawing 


10.124 


.785  (6.4375«-5  .3437*  ) 
104  Ibs/sq.in. 


Oil  pressure  in  recuperator  =  576  +  1350 

A  .      Initial  Final 

-  -  -  =  963  Ibs/Sq.in 

I 

Oil  pressure  in  recoil 

2222+1670 

=  -     »   1946  Ibs/sq.in. 
2 

Recuperator   stuffing  box  diam.   =   2.169  length  of 
contact  (dermatine    )=.787 

Friction   -    .  75x2  .169x3  .  14"  (963  +104)x.09x.  787»375. 
Recoil   stuffing   box  diam.   =4.728   Length  of  contact 

«    .787 

Friction  «  .  73x4.  728*3.  14*  (1946+104)*  .09x  .787=1572 
Total  stuffing  box  friction  =  Recoil  stuffing  box 

friction  +  Recuperator 
stuffing  box  friction 

»  1572+375*1947  Ibs.  Total  stuffing  box  friction. 
Total  friction  »  guide  *  stuffing  box. 

-  8450 
_____   *  1947 


10397  Ibs. 
Calculation  of  Throttling  Areas  . 


545 


2[K-  -(b+X-2Xr)](b-x) 

« 
C  A 


13.2  /K-pa-rRt+Wr   sin  0 

But 

'[K-  7(b+X-2Xr)J(b-x) 

•»    V, 


C  A*   Vx 
W, 


13.2/K-pa-Rt+Wr   sin  0 


Pa  =  Pai  x  Ar^PP1"0*)3*11^*4!  pressure  x  ef- 

fective area  of  piston 
=  576  x  35.766  =  20600 
C  =  1.39  (constant) 
A  =  35.766 

Hr  sin  0  =  15790  x  .0848  =  15550 
Rt=guide  friction  +  stuffing  box  friction  = 

10,000  Its. 

K  =  155000 

Vx=  take  the  values  as  calculated  for  vol.  curve, 
From  calculations: 

when  x  =  3.57" 

V  =  36.449ft.sec. 

wx=  .061  x  36.449  =  2.223  x  2  =  4.446 
When  x  »  10.39in. 

V  =  42.118ft.sec. 

wx=    .061    x    42.118  -   2.5691   x    2    *  5.1382 
When  x  =   10.94in. 

V   =   41.7ftsec. 

*rx*.061    x    41.7  =   2.5437   x   2   =  5.0874 


When   x  *   1.5ft.or   18in. 

Vx   =  37.4ftsec.  3 

W  L39   x   3S.7662 


x        13.2  /154725  -  20600-10397+15550 

.061x37.4=2.2814   sq.in.*2    rods=4.5628 


546 


l& 


^ 

<s 


s 
fe 


I 


X, 


Y 


D/> 


t 


•* 5- 


3&W6? 


§ *- 


_ 


547 


546 


When  x 
V, 


2ft.or  24in. 
•33.6 
«    .061  x  33.6 


2.05    sq.in.    x    2    =   4.10 


When  x  -  3ft.or  361  n. 

Vx-  22.8 

wx». 061x22. 8-1. 28  sq.in.x2«2.76 
When  x  «  3.75ftor  44.8in£total   recoil) 

w  -   0 


Comparison   of  Throttling   Areas. 

I  no  b  a  a     Recoil 

Calculated     Area     of 

Orifice     (2     rod.) 

Frenoh     Value 

3.5-7 

4.«46 

4.  413 

10.39 

9.  13** 

b.  129 

10.94 

5.08I74 

5.084 

18. 

4.5628 

4.54 

24. 

4.  10 

4.08 

36. 

2.76 

2.  69 

44.8 

0. 

0. 

SCHNEIDER  COUNTBR  RBCOIL. 


The  counter  recoil  is  divided  into  three 
periods: 

(1)  The  accelerating  period,  the 

:        counter  regulation  being  controlled 
by  a  constant  orifice  through  the 
buffer  in  the  recoil  rod. 

(2)  The  retardation  period,  the  count- 
er recoil  regulation  being  controlled 
by  a  variable  throttling  orifice 
through  the  buffer  head. 


549 


(3)     A  constant  orifice  period  at  the 
end  of  recoil,  the  throttling  orifice 
being  very  small  and  the  displacement 
a  very  small  part  of  the  recoil. 
The  displacements  corresponding  to  (1),  (2) 
and  (3)  are  1Q,  lb  and  1Q  respectively. 

Counter  Recoil  Data. 


Length  of  constant  orifice  1Q  =  31.3  inches 

Length  of  variable  orifice  1^  =  7.85  inches 
Length  of  constant  orifice 

at  end  of  recoil  1  =  5.68  inches 


b  =  Total  c  'recoil        44.83  inches 

Constant  orifice  period        0.7  b 
Variable  orifice  period        0.175  b 
Constant  orifice  at  end  of 

c  'recoil  0.125  b 

where  b  »  length  of  recoil 

There  being  2  recoil  brakes,  we  have  for  the 
buffer  reaction: 

.  «»•*;»; 

B-  * 


where   A^   =  area  of   one  buffer          =   9.859 
aQ  »   area  of  constant  buffer 

orifice  =   .0664  sq.in. 

ao  =  area  of  constant 

buffer  orifice  at  end 
of  recoil  -   .022  sq.in. 

Considering  the  c  'recoil  at  horizontal  elevation, 
during  the  constant  orifice  period,  we  have 


•o  »o  2-3026 

where   A  =   load  on  air  -  friction 
=   Py   -  I  R 


550 


and  C0  « 


ZR 


2c  Ab 

^•^•MM 

175 

6290  Ibs. 
490 


2.78  x  940 
175 


15 


A  x 

Total 

L                       OT* 

2o(a«-*M 

V 

Buffer 

in. 

*    in. 

»* 

2.3    »»* 

force 
t 

.  A 

.  4 

397oo      3415V* 

.  198 

1.16 

4550. 

.  8 

1.  2 

37860 

.386 

2.67 

243OO. 

.  8 

2. 

36860 

.386 

3.05 

31700. 

1. 

3. 

35810 

.493 

3.3 

37100. 

1. 

4. 

34910 

.493 

3.23 

35500. 

1. 

5. 

33910 

.  493 

3.18 

34400. 

1. 

6. 

33160 

.493 

3.  13 

33500. 

1. 

7. 

32510 

.493 

3.09 

32900. 

5. 

12. 

28710 

2.5 

2.91 

29000. 

5. 

17. 

25710 

2.5 

2.745 

25720. 

5. 

22. 

22710 

2.5 

2.5 

22700. 

5* 

27. 

20710 

2.5 

2.46 

20750. 

4.3 

31.3 

18910 

2.13 

2.36 

18950. 

Beginning  of  Variable  Orifice. 
After  this  period  the  unbalanced  force  was 
assumed  constant. 

*  ib  *  r  M*o  -  *;> 

.3-. 25 


0  »  245  (- 


.655 


-)  =  1880.  #  =  unbalanced  force 


175  ** 


551 


I  n  o  b  •  • 
Reooil. 

x  in 

1880  x 

1300-1800 

x 
V   Total  lb«. 
Buf  f  »r 

245 

32.3 

33.3 

.0833 
.  1666 

157. 
360. 

4.61? 

3.94 

2.  16 

1.99 

20500 

20200 

34.3 

35*3 
36.3 
37.3 
39.  15 

.25 
•  333 

.416 

•  5 
.656 

47o 
627. 

780. 
940. 
1230. 

3-38 

4.75 
2.  12 
1.47 
.286 

1.  84 

1.66 

1.45 

1.  215 
.52 

9900 
19500 
190OO 
18500 
18000 

40. 

.5 

I750u 

17000 

42. 

44.  83 

•  5 
.0 

16500 

16000 

3  o  i  n  t   X  Foot 

45  80  z 

323.5-  4580x 
^215 

V.I 

Le  »  d 

X 

458OX 

245 

f.«. 

on 

Air 

1 

.0833 

382 

2833 

11. 

5 

3. 

39 

28400 

116 

2 

.1666 

762 

2453 

10. 

3. 

16 

27600 

108 

3 

.25 

1145 

2070 

8. 

45 

2. 

91 

26800 

.  102 

4 

•  333 

1525 

1690 

6. 

9 

2. 

63 

26000 

.092 

5 

.  416 

1900 

1315 

5. 

38 

2. 

32 

24600 

.083 

6 

.5 

2290 

925 

3. 

78 

1. 

945 

24400 

.065 

7 

.656 

3000 

215 

8. 

76 

* 

94 

21000 

.044 

ftHKttlDgB 


X  •  any  interval 

0  •  unbalanced  force 

Mr  =  mass  recoiling  parts 

Vo  *  max.  velocity  of  o 'recoil 

Vx  »  velocity  at  any  point 

Vx  *  velocity  at  beginning  of  period  le 

1  >  length  of  constant  orifice  period  in  feet 


552 


1^  s  length  of  variable  orifice  period  in  feet 
lc  =  length  of  final  period  for  c 'recoil  in 

feet 

Pa  =  load  on  air  in  Ibs. 
R  »  Total  friction 

2K*AV 

=   total  "buffer   force 

175  W» 

b   =   length  of  c'recoil    (ft) 

Period   1Q 
2K2A»Va 

Pa  -  R  -  =   0       acceleration  »   0 

175A» 

Assume  velocity  of  3.5  ft.  per  second  and  solve  for 
orifice  W, 


Period  lc 

pa  -  R =  0 

175  W* 

Assume  velocity  of  1  ft.  per  sec.  and  solve  for 
*x 

Period  lu 


x  = 


Knowing  V   solve  for  Kx  for  various  points 


553 


175 


Pa  =  R  Solve  for  Wx 


ST.   CHAMOND  RECOIL. 

ST.  CHAMOND  RECOIL     The  type  of  St. diamond  brake 
SYSTEM.  here  discussed,  consists  of  three 

cylinders;  a  hydraulic  brake 
cylinder,  a  recuperator  cylinder 
containing  the  floating  piston 

which  separates  the  air  and  oil,  together  with  a 
regulator  valve  for  throttling  the  oil  between  the 
hydraulic  and  recuperator  cylinders,  a  third  cylinder 
serving  as  a  part  of  the  air  reservoir  and  therefore 
communicating  with  the  recuperator  cylinder  air 
volume,  the  remainder  of  the  third  cylinder  being 
used  for  storing  oil  for  the  brake  mechanism. 

One  of  the  peculiar  features  of  this  type  is 
the  regulated  spring  valve  where  the  main  throttling 
occurs.    The  valve  functions  somewhat  as  a  pressure 

regulator  or  governor,  since  if  the  pressure  falls, 
the  spring  reduces  the  valve  opening  tnereby  in- 
creasing the  throttling  drop  and  the  pressure  in 
the  hydraulic  cylinder.   The  pressure  in  the  recoil 
cylinder, (i .e.  the  hydraulic  pressure)  is  the  sum  of 
the  air  pressure,  plus  ttie  floating  piston  friction 
drop,  plus  the  throttling  drop  through  the  regulator 
valve.    At  short  recoil  the  air  pressure  is 
necessarily  small  compared  with  the  throttling  drop. 
The  resistance  to  recoil  is  large  and  therefore  the 
recoil  pressure  large.   This  requires  a  large  throttling 
drop  and  the  air  pressure  becomes  necessarily  small 
compared  with  the  throttling  drop.   The  large 
throttling  drop  requires  a  very  small  valve  opening, 
with  a  large  pressure  reaction  against  the  valve. 
To  balance  this  reaction  a  very  stiff  spring  is  re- 
quired.   Such  spring  characteristics  have  been  ad- 


554 


mirably  met  oy  the  use  of  Belleville  washers.   At 
long  recoil  the  resistance  to  recoil  ia  small,  there- 
fore the  throttling  drop  is  small,  requiring  a  large 
orifice  area.   Since  the  pressure  in  the  recoil 
cylinder  is  small  together  with  a  large  orifice 
opening,  a  weak  spring  with  large  deflection  is 
desirable.   Such  spring  characteristics  are  best 
met  with  an  ordinary  spiral  spring.   Hence,  at  long 
recoil,  low  elevation,  a  spiral  spring  functions 
alone,  while  at  short  recoil  maximum  elevation  the  belle- 
ville  and   spiral  spring  function  in  parallel.  The 
regulator  ia  so  designed  that  at  low  elevation  only 
the  spiral  spring  functions. 

To  modulate  or  regulate  the  velocity  of  count- 
er recoil  to  a  low  velocity,  the  pressure  in  the 
recoil  cylinder  is  lowered  just  sufficiently  to 

balance  the  total  friction  during  counter  recoil. 
At  the  end  of  counter  recoil  the  recoil  cylinder 
pressure  is  reduced  to  zero  and  the  recoiling  mass 
is  brought  to  rest  by  the  total  friction  alone. 
To  reduce  the  pressure  during  the  first  part  of 
counter  recoil  throttling  through  a  constant  orifice 
la  effected  in  a  separate  passage  way  or  channel 
leading  from  the  recuperator  to  the  recoil  cylinder. 
At  the  end  of  counter  recoil  additional  throttling 
around  a  buffer  rod  and  its  chamber,  is  effected 
reducing  the  pressure  in  the  recoil  cylinder  to 
zero  or  nearly  so. 

DESCRIPTION  OF  THE     Referring  to  figure  (10)  is 
OPERATION  OF  THE  ST.  shown  a  schematic  diagram  of 
CHAMOND  RECOIL.      the  operation  of  the  St. 

Cnamond  recoil  system  for  both 
recoil  and  counter  recoil. 
Recoil:-   During  the  recoil  a  flow  or  stream  of  oil 

passes  by  the  regulator  valve  from  the  hydraulic 
to  the  recuperator  (oil  side)cylinder.  The  pres- 
sure p  of  the  oil  against  the  recoil  piston  is  re- 


555 


1 


556 


RECO/L  REGULATOR 


Fig- 6 


557 


duced  by  throttling  through  the  regulator  to  a  pres- 
sure (pa)  against  tne  oil  side  of  the  floating  pis- 
ton. Due  to  the  friction  of  tbe  floating  piston  the 
air  pressure  pa  is  less  than  tbe  pressure  on  the 
oil  side  of  the  floating  piston  p^.   The  tension  in 
the  recoil  rod  is  balanced  by  tne  total  pressure  on 
the  recoil  piston  plus  the  hydraulic  piston  friction 
plus  the  stuffing  box  friction  in  the  recoil  cylinder. 
The  valve  in  the  counter  recoil  orifice  remains 
closed  during  the  recoil. 

REGULATOR  VALVE.     The  throttling  during  the  recoil 
is  controlled  by  tbe  regulator  valve. 
See  figure  (11).   The  regulator  valve 
consists   of  two  parts:  an  upper  stem 
and  the  lower  valve  stew.  The  lower 
valve  stem  is  seated  very  carefully  on  a  circular 
seat  at  the  top  of  the  entrance  channel.   As  the  valve 
lifts,  the  throttling  area  becomes  the  vertical  cir- 
cumferential area  between  the  valve  and  its  seat. 
The  spiral  spring  reacts  on  the  lower  valve  stem. 
The  Belleville  washers  at  the  top  of  the  upper  stem, 
react  only  on  that  valve  stem.   The  upper  stem  rests 
in  a  valve  box  or  housing.   To  move  the  upper  valve 
stem  (other  than  the  slight  deflection  possibly 

compressing  the  Bellevilles)  the  whole  housing  or 
valve  box  is  moved  by  a  cam  as  shown  in  diagram. 
The  diameters  of  the  upper  part  of  tne  lower  valve 
stem  and  the  lower  part  of  the  upper  stem,  (that  is 
the  diameter  of  the  stems  of  the  regulator  valve,) 
are  tbe  same.   At  short  recoil  the  reaction  of  the 
Belleville  on  the  upper  stem  is  transmitted  by  the 
mutual  reaction  between  the  upper  and  lower  stems 
at  their  surface  of  contact. 

The  valve  opening  and  consequent  throttling 
drop  of  pressure  depends  upon  the  deflection  of  the 
spiral  springs  or  Belleville  washers,  the  spring 
reaction  balancing  the  hydraulic  reaction  on  the 
valve.   Neglecting  the  small  dynamic  reaction,  the 


558 


hydraulic  reaction  on  the  valve  is  fhe  product  of 
the  intensity  of  pressure  in  the  recoil  cylinder 
and  the  base  of  the  regulator  valve,  minus  the 
product  of  the  intensity  of  pressure  in  the  re- 
cuperator cylinder  and  the  effective  area  on  the 
upper  part  of  the  regulator  valve.  At  long  re- 
coil, since  the  loner  valve  stem  comes  in  con- 
tact with  the  upper  stem,  the  effective  area  on 
the  upper  part  of  the  valve  is  obviously  equal  to 
the  area  at  the  base  of  the  valve.   Hence  the 
hydraulic  reaction  at  long  recoil  is  merely  the 
product  of  the  difference  in  pressures  between 
the  recoil  and  recuperator  cylinders  and  the 
area  at  the  base  of  the  valve.   At  short  recoil  the 
upper  stem  of  the  regulator  is  brought  down  by  the 
cam  at  its  top,  until  its  lower  surface  is  in  con- 
tact with  the  top  surface  of  the  lower  valve  stem. 
The  effective  area,  therefore,  on  the  upper  part 
of  the  regulator  valve  equals  the  difference  in 
areas  between  the  area  at  the  base  of  the  lower 
valve  stem  and  the  area  at  the  upper  end  of  the 
lower  valve  stem,  or  the  area  of  the  upper  valve 
stem;  the  two  latter  being  always  equal.   Hence 
the  hydraulic  reaction  at  short  (or  intermediate 
recoil  for  the  greater  part  of  recoil)  equals  the 
product  of  the  recoil  intensity  of  pressure  and 
the  base  of  the  valve,  minus  the  product  of  the 
recuperator  intensity  of  pressure  and  the  difference 
in  areas  between  the  base  and  middle  stem  of  the 
valve,  when  upper  and  lower  stems  are  in  contact. 
At  long  recoil  the  hydraulic  reaction  is  balanced 
above  by  the  spiral  spring  reaction.   At  short 
or  intermediate  recoil  the  hydraulic  reaction  is 
balanced  by  the  combined  reaction  of  the  Belleville 
washers  and  the  spiral  spring  though  the  latter  is 
negligible  compared  with  the  former. 
COUNTER  RECOIL.         The  regulator  valve  it 
closed  during  counter  recoil. 
The  oil  flow  during  counter  recoil,  therefore,  is 

different  from  that  in  recoil.  The  valve  is  seated, 


559 


but  the  oil  is  allowed  to  pass  through  a  very  small 
hole  in  its  center.  This  orifice  is  constant  through- 
out the  whole  of  counter  recoil.   There  is  another 
channel  for  the  oil  leading  from  the  bottom  of  the 
buffer  chamber  in  the  regulator  body.  This  oil 
passes  through  a  ball  valve.   As  the  floating  pis- 
ton returns  to  its  initial  position  at  the  end  of 
counter  recoil,  the  regulator  rod  enters  the  buffer 
cavity,  thus  obstructing  entrance  of  oil  to  this 
cavity.  This  rod  is  tapered  so  that  when  it  has 
fully  entered  the  cavity  there  is  no  clearance 
between  the  rod  and  the  entrance,  and  tne  oil  in 
returning  to  the  recoil  cylinder  nust  all  pass 
through  the  central  opening  in  the  valve.  By 
neans  of  this  regulation  it  is  possible  to  allow 
the  gun  to  return  to  its  "in  battery"  position 
quickly,  but  its  final  movement  is  so  controlled 
that  there  is  no  ehock. 

The  throttling  areas  in  the  counter  recoil 
channels  are  so  designed  as  to  cause  sufficient 
throttling  to  lower  the  pressure  in  the  recoil 
cylinder  that  it  nay  practically  balance  the  total 
friction,  during  the  counter  recoil.   At  the  end 
of  counter  recoil  this  friction  alone  brings  tne 
recoiling  mass  to  rest  when  it  reaches  the  battery 
position. 

GENERAL  THEORY  OF  THE     Figure  (11)  shows  the 
ST.  CHAMOND  BRAKE.     regulator  valve  stem  for 

both  long  and  short  re- 
coil. 
Let  Rv  »  reaction  on  base  of  throttling  valve 

p  *  intensity  of  pressure  ia  recoil  cylinder 
pa  *  intensity  of  air  pressure. 

pa  •  intensity  of  pressure  in  recuperator 

cylinder  (i.  «.  on  oil  side  of  floating 
piston) 

a  •  entrance  area  of  valve  or  effective  area 
at  base  of  valve. 


560 


a  =  area  of  valve  stem 

Sb  *  spring  constant  of  belleville  washer 
Ss  =  spring  constant  of  spiral  springs 
C  =  effective  circumference  at  base  valve 
h  =  lift  of  valve  from  initial  opening 
h  »  the  initial  compression  of  the  spiral 

3 

valve  spring  at  initial  opening 
hb  *  the  initial  compression  of  be  Seville 

washer  at  initial  opening 
w  =  throttling  area 
v  =  velocity  of  flow  through  entrance  area 

"a" 

V  =»  velocity  of  recoil 
A  =  effective  area  of  recoil  piston 
d  =  density  of  oil 

The  hydraulic  reaction,  at  long  recoil  becomes 

i 
Rv  -  paa,  and  at  short  recoil,  we  have,  the  value 

Ry  -  pa(a-at).   The  belleville  washer  reaction, 

oecomes  Rb  =  Sb(hb+h)  and  the  spiral  spring  reaction, 

becoraes,  Rs  =  Ss(hs+h).  Hence  at  long  recoil,  we 

nave  Ry-paa  =  SS  (hs  +  h)  +  F    (1) 

and  at  short  recoil,  we  find 

Kv-pa(a-at)*Ss(hs+h)+Sb(hb+h)+F   (2) 

that  is,  Rv-p^(a-at)=Sshs+Sbhb  +h(Ss+Sb)+F 

Now  at  intermediate  recoil  the  upper  valve  stem 
is  separated  from  the  lower  valve  stem  by  a  distance 
ho  when  the  latter  is  just  about  to  leave  its  seat. 
If  hQ  is  tne  separation  between  the  two  stems  before 
recoil  and  if  e  =  the  initial  lift  of  lower  valve 
stem  required  to  clear  the  valve,  then  ho  =hQ  =  e 
Hence  at  intermediate  recoil,  we  have 
Rv-pa(a-at)=S8(hs+h)+Sb(n-n0+hb)+F,  that  is 
Rv-pa(a-at)=Sshs*Sb(hb-h0)+h(Ss+Sb)+F       (3) 
Where  F  is  the  valve  stem  friction  and  will  be 
neglected, let 

C0»Ssh8  and  CQ  =Sghs+Sbhb 

co*ssns  *3b(hb-ho> 


561 


The  reaction  against  the  base  of  throttling  valve, 
in  terms  of  the  pressure  at  the  entrance  to  valve, 

becomes,  2 

Rv-p  a  =  -^-  (4) 

g 

where  PJ  =  the  pressure  at  a  mid  section  in  the 

entrance  channel  of  the  valve. 
Further       2 

—  *  —  =  E  +  ht          (5) 
d    2g   d 

Neglecting  the  friction  and  accelerating  head,  ht 
as  snail  we  have,  therefore 

dv* 
p  =  p  -  wnich  gives  the  pressure  in  the 

"&   entrance  channel  in  terms  of  the 
recoil  pressure  (i.e.  the  pressure  against  the  hy- 
draulic piston)  hence 

dav         dav    dav 
Rv-Pta+  -J—    Pa    2g     g 


or 

dav         dv  . 

R=pa  +  =  (p  +  )a        (6) 

2g          2g 
Therefore  at  long  recoil,  we  have 

dv 

(p+  )a=C   +Ssh  +  paa 

2g 

and  at  short  recoil   we  have, 

dv* 

(p+  )a=C0+(Ss+Sb)h+PaU-at)     (8) 


dv* 
(p+  )a=C0+(Ss+Sb)h+pa(a~a  )     (9) 

2g 

Considering  now  the  main  throttling  through 
the  circumferential  section,  around  the  effective 


562 


circumference  of  the  valve,  we  have,  for  the  ef- 
fective throttling  area, 

oh  1 

w  »  —    where  K0  »   '      Contraction  factor  of 

Ko  0-775   orifice, 

tbe  corresponding  pressure  drop  through  the 
valve  becomes, 


K!A"V' 


p  - 


175(c«h«) 


Further  av  »  AV.  hence  V*=(T)*  V* 


hence     lp+  •$£(£) 

b  * long  recoil  (10) 


Sg+Sb                                  recoil    (11) 

and 
ll 

f  $-(-)*   v*        _c«_    if 

S8+  Sjj                                  mediate  re- 

coil  (12) 

Considering  only  the  main  throttling,  or 
ratber  designing  the  recoil  flow  channels  to  have 
throttling  as  compared  with  tbe  throttling  through 
the  regulator  valve,  we  have 

P  *  P  *  Pa  hence  we  have  the  three  fundamental 
equations  for  the  recoil  pressure,  in  terms  of  the 
velocity  of  recoil  and  tbe  pressure  in  tbe  re- 
cuperator cylinder: 

KSS8AV 
p,  -  -  -  -  +pa   (13) 

175Ca[(p+  -(-)"  —  ]a-C-P4a)'        ** 


2g  a   144  recoil 


d   A   V* 

175C*[(p+  — (-)*  )a-C'-pa(a-at)a     at 

2g  a   144  short 

recoil. 


AV 


+  pa  (15) 


175C*[(p+  —  (-)*  —  a-C0'-pa(a-a  )]'    _ 
2g  a   144 

mediate 

recoil. 

where  the  units  are  obviously,  p'a  and  p  in  Ibs. 
per  sq.in. 

V  in  ft.  per  sec. 

Aiaiand  at  in  sq.  in. 

d  in  Ibs.  per  cu.ft. 

If  further  J0  *  I  »  a~  *  144  then  «9uati°n8  (13), 

(14)  and  (15)  reduce 
to  the  simpler  form 


-  :;  -  ;  -  z—   (16)  at  long  re- 
175C  [(p-pa)a+J0V  -C0]         coil 


p_p  „  -  .  -  ..  - 
175  C  [(p-Pa)a+Paa+J0V  -CQ] 


recoil 
K*(Ss+Sb)A*V* 


;;  —   (18)  at  in- 
175C  [(P-Pa)a+P;a+J0V  -CQ"]      ter.ediate 

recoil 

To  compute  p  for  any  given  displacement  and  cor- 
responding recuperator  pressure  and  recoil  velocity, 
we  find  the  solution  in  the  form  of  a  cubic  equation 

The  solution  is  as  follows: 
From  equations  (16),  (17)  and  (18), 

KV.AV  ' 


175Caa» 
p~pa  " 


564 


or 


P-Pa 


VV* 


175C8a2 


lot 

B  ,  K  A'V'S*   or 

175C«a*  175  C«a* 

p-Pa  *  z 

Jovico      PaW'-Co 

and  or  =  a 

a  a 

Then  from  the  above  equations,  we  have 


or   B=Z   +2Z  m+Zm 

To  eliminate   the     2nd  degree   term,    substitute, 

2  a      *        4  4 

Z  »   X  -  r  m          hence    Z  -X     -  -  •  X   +  -  n 

O  39 

and  , 

3  3  9  •*  9  H  * 

Z          —       V^      O      m      Vo.  »  V 

A  G     ifl     A     ~  Ql         A"1"     ~" T      ID 

Expanding,   we   find 

4      »  8      s    x     ,,a         82.. 


B   =  X 

3  2?  3 

*  8-  "'  *  ,8x  -  f  .' 

=   X3    -  -  m*X  -  —  m8 

Further   let  N3    *  —  +  -7  then   X*   -  -  m*X-N%3»  0 
^7       m°  3 

Solving  by  Cardan's  method 


^ 

—    +   /  —   —  •»    /  —  —  /  —  —   •          )    m 

2  4  730  2  4  730 


565 


2 

X    a   Z 

+      ~~       Hi 

+   - 

3 

a        3 

hence 

* 

N8 

.                                           3    / 

/i!        _1_ 

2^ 

p  »(/ 

2      * 

4~    "   730   +        2~    "   ' 

4      "  730 

"3} 

During  the  greater  part  of  recoil  except  at  the 
very  beginning  and  towards  the  Very  end  of  re- 
coil it  has  been  found  by  actual  calculations,  that 

the  term  —*•-  becomes  negligible  in  comparison  with 
730  ^  e 

—  and  nay  be  omitted  without  appreciable 
4 

error. 

The  above  equation,  therefore  reduces  to  the 

simple  form 

2 
p<=  m(N  -  g)+  pa 

Another  and  far  simpler  method  for  the  com- 
putation of  p  established  by  Mr.  McVey,  consists 
in  the  construction  of  a  table,  with  assumed  values 

of  P  ~  Pa. 

The  table  is  based  on  the  two  following  equations 
(neglecting  dynamic  bead  as  small) 

(p-pa>A*PaAi  s  co+(ss+sb)h  short  recoil 

(a) 
(p-pa)A=C0+Ssh  long  recoil 

pa  A«Ya 
and  p-pl  -  175C,h, 

If  ve  assume  a  mean  air  pressure  throughout,  (the 
error  thus  introduced  having  been  found  small), 
we  have 


or  = 


ss 


Assuming  a  series  of  values  of  (p-pa)  «e  obtain  a 
series  of  values  for  h,  now  from  (b) 


566 


567 


568 


569 


from  which  a  series  of  values  can  be  established 
for  corresponding  values  of  p-paand  h. 

Knowing  the  retarded  velocity  for  any  given 
point,  the  corresponding  value  of  (p-pa)  and  h  can 
be  picked  from  the  table  and  knowing  pa  for  the  given 
point  in  the  recoil,  the  recoil  pressure  p  is  ob- 
tained. 

It  is  to  be  noticed,  that  substitution  of  (b ) 
in  (a)  gives  a  cubic  with  a  second  degree  term,  as 
before.   Thus  no  direct  simple  solution  is  possible. 
The  table  method  is  recommended  even  for  short  re- 
coil since  the  error  introduced  by  assuming  the  air 
constant  is  relatively  snail. 

GENERAL  PROCEDURE  FOR     Due  to  the  complexity  of 
CALCULATION  OF  RECOIL,   the  general  equation  of  re- 
coil no  mathematical  solution 
is  possible,  except  by  ex- 
panding into  a  series.   Such 

a  solution  of  a  recoil  equation  is  known  as  the 
"point  by  point"  method  and  has  been  used  before 
in  this  text. 

The  object  of  actual  computation  of  rec-oil 
curves  for  a  given  type  of  mount  is  to  ascertain 
the  ratio  of  the  peak  to  the  average  resistance  to 
recoil  at  maximum  and  zero  elevation.   The  average 
resistance  may  be  readily  obtained  in  the  preliminary 
layout  of  a  design  and  knowing  the  peak  ratio  for 
a  given  type  of  mount,  enables  the  peak  resistance 
to  be  obtained  and  the  consequent  stresses  in  the 
carriage.   Let 

Vf0  =  free  recoil  velocity  at  point  "n"  (i.e. 
the  velocity  generated  in  the  recoil- 
ing mass  by  the  powder  pressure). 
Vrn  »  corresponding  retarded  recoil  velocity 
Rn  =  total  friction,  stuffing  box  and  guide 

friction. 

$  =  angle  of  elevation  of  gun. 
Now,  the  end  pressures  at  the  beginning  and  end  of 


570 


<*JU- 

recoil,   becomes   p-pi   *  —  for 

ft 

a  -at   C0 
p*  Pa(  -  )*  —  for  short  recoil 

A         & 

Since  0  =  0  at  long  recoil,  we  have  for  the  re- 
sistance to  recoil,  K  »  pnA  +Rn  for  long  recoil 
K  »  pnA+Rn-Wrsin0  for  short  or  intermediate  recoil 

Long  recoil: 

For  1st  point  long  recoil, 


Pao   +          +  R 


o 


Vrt    "  vft 


and   knowing    yri 

3 


,  A*  A*   i   A*   A*  i   2. 

•   a   (   /  — +  /  —  —  — —  +  /  —  —  J  —  —  — —  —  _\m+Dl 

4     4    730     2      4    730   3   Pa, 


m  = 


175C2a2  a 

For  2nd  point  long  recoil, 


and  knowing  Vra 


730 


{>AtVr.S8  J0V«a    -C0 

B  »  V--   ;  m  =     f   — 

175Caa»  a 

After  a  very  few  intervals  the  valve  opens  sufficient 
ly  so  that  the  term 


571 


— rr  may  be  omitted,  then  for  point  "n"  at  long  re- 

I  o(j 

coil, 

(pmA+Rn) 

Vn-V<vrrrvrm> ' At      and   knowing   Vn 

mr 
2 

Pn  -  m(N  -  -)+pa   Obviously  after  the  powder  pres- 
sure period  Vfn-Vfm  =  0  and  we 
have  the  simple  dynamic  equation  of  recoil. 
Short  Recoil: 


The  procedure   for   the   calculation  of   the 
velocity   and  pressure   curves   for   short   recoil   is 
exactly   similar   as-    for     long   recoil. 
For   1st  point   short   recoil, 

P«      (— —  )  +  —  +R  -Wrsin0 

c*  1  a  a  1         * 

»r,  "f  ,  -< ~ T5 >   4  * 


and   knowing   V_ 
a  *  i 


,    /N8 /N«~      1         /N  A*~      1          2. 

a(     /    — +    /    —    —  +    /    —    —    J    —    —    —    —    — )m+Dl 

1  2  4  730          2  4          730        3'      Pat 


P 
where 


175C2aa  a 

for  2nd  period  short  recoil, 

p'A+R  -W  sin  0 

V     ~v     =vf  ~vf     -   C  - 
ii 


and  knowing  Vr2  ,  p2  can  be  obtained  by  a  solution 
of  the  previous  cubic  equation.   The  greater 
number  of  points  of  recoil  excepting  a  few  points 
at  the  beginning  and  end  may  be  solved  with  suf- 
ficient accuracy  by  the  expression, 

2 
pn=m(N-  -gO+Pa  "nere,  as  before, 


572 


V^-—  K«A«V«(Ss+Sb)» 

N  -  / —  +  —  B  »  

27   •"  175C»a« 

and 


Calculation  from  constructed  table  of  (p~Pa)»  h, 
and  V~ 

The  procedure  here  is  exactly  similar  as  above; 
each  preceding  interval  establishes  a  new  retarded 
velocity  which  from  the  table  establishes  a  new 
recoil  pressure.   This  recoil  pressure  substituted 
in  the  dynamic  equation  in  turn  establishes  the  re- 
tarded velocity  at  the  end  of  the  interval  under 
consideration. 

Judgment  must  be  aged  in  the  proper  increments 
of  time  to  be  used.   The  closer  the  intervals 
the  more  accurate  the  velocity  and  pressure  curves. 
At  the  beginning  and  end, the  time  intervals  should 
oe  taken  smaller.   During  the  major  part  of  re- 
coil the  time  intervals  can  be  fairly  large.   As 
a  check  during  the  powder  period  the  retarded 
velocity  should  be  roughly  0.9  of  the  free  velocity 
of  recoil. 

CALCULATION  OP  THE  VARIOUS     In  the  calculation 
FRICTION  COMPONENTS  DURING   of  the  vertical  pres- 
RECOIL.  sure  and  retarded 

velocity  curves  for 
the  St.  Chamond  brake, 

the  frictions  vary  as  a  function  of  the  pressure. 
At  long  recoil  the  pressure  variation  is  small 
and  we  are  not  in  great  error  in  assuming  constant 
friction:  with  short  recoil,  however,  a  peak  value 
is  obtained  and  with  it  a  change  in  friction. 

The  frictional  resistance  opposing  recoil  are: 
(1)     Guide  friction  which  is  function 
of  the  total  pull. 


573 


(2)  Stuffing  box  friction  which  is 
a  function  of  the  recoil  pres- 
sure. 

(3)  Recoil  piston  friction  which 
also  is  a  function  of  the  recoil 
pressure. 

(1)     The  guide  friction  during  recoil  has  been 
previously  expressed  by  the  following  equations: 

2(pbe+Bb)+Wrcos0N 
R     =  -  n 

*  n 

where  pb  is  the  powder  reaction  on  the  breech 
c  is  the  perpendicular  distance  between 

the  axis  of  the  bore  and  a  line  through 
the  center  of  gravity  of  the  recoiling 
parts  parallel  to  the  axis  of  the  bore. 

B  =  pA,  the  hydraulic  reaction  of  the  re- 
coil piston 

n  =  coefficient  of  friction,  from  0.15  to  0. 
20 

"b  *  distance  down  of  the  line  of  pull  from 
the  center  of  gravity  of  recoiling  parts 


where  XA  and  yt  are  the  coordinates  of  the  front 

clip  reaction  and  x^  and  yg  are  the  coordinates 
of  the  rear  clip  reaction  having  axis  and  origin 
through  the  center  of  gravity  of  the  recoiling  parts 

Considering  the  somewhat  inaccuracy  of  a 
"point  by  point"  method  of  computation,  it  is  be- 
lieved the  following  formula  is  sufficiently 

accurate,     2nBb  +  nWrcos  0(xt-xa) 
R  =  -  - 

C  -  2nr 
or  when  *~x   *s  small, 


where  r  is  the  mean  distance  from  the  center  of 
gravity  of  the  recoiling  parts  to  fhe  line  of 
action  of  the  guide  frictions. 


574 


(2)     The  packing  friction  formulas  have  been 
already  considered  in  more  or  less  detail  in 
Chapter  VIII.   The  stuffing  box  friction, 

R8  »  c^  +  c;  P 

where 

c 


t 


Ct-  ndr(bf+af+atft) 
(3)     The  hydraulic  piston  friction, 


Rp=C"  *  C 


ndp0(bf+afi) 


From  the  above  formulae 

p  *  recoil  pressure  in  Ibs.  per  sq.in. 
P0=intensity  of  pressure  caused  by  Bellevilles 

or  packing  springs  in  Ibs.  per  sq.in 
Rb  *  belleville  or  packing  spring  reaction  on 

annular  area  of  packing  spring  at  as- 

sembled load  in  Ibs. 
dr  *  diam.  of  piston  rod  in  inches. 
do  =  outer  diam.  of  stuffing  box  packing 

ring  in  inches. 

d  *  diam.  of  recoil  cylinder  in  inches. 
di=inner  diam.  of  piston  packing  ring  in 

inches. 
b  =  width  of  leather  contact  of  packing  in 

inches. 
f  *  corresponding  coefficient  of  friction  * 


-  -  silver  contact  of  flap  of  one  flange  of 

packing  ring  in  inches. 

ft»  coefficient  of  silver  friction  »  .09 
Then  po  becomes, 


for  (Dp 


575 


In  summing  up  the  component  frictions,  we 
have 

nWrcos  0(xt-xf) 


C-2nr 


RS  =  c;  +  ct'P 

Rp  -  ct"  +  c;  P 

hence  R   -    (C»*C|[*Kt  )+(CJ    +  C;1  +K2)   p 

»  c4+csp 

showing  the  total  friction  resisting  recoil  is  a 
linear  function  of  the  pressure  in  the  recoil 
cylinder. 
Floating  piston: 

The  oil  pressure  in  the  recuperator  cylinder 
during  the  recoil  is  greater  than  the  air  pres- 
sure by  the  drop  of  pressure  caused  by  the  float- 
ing piston  friction.   In  the  previous  recoil 
equations,  the  recuperator  oil  pressure  has  been 
used  in  place  of  the  air  pressure.  To  compute 
this  pressure  knowing  the  air  pressure,  it  is  only 
necessary  to  compute  the  floating  piston  friction 
drop.   In  the  discussion  of  the  floating  piston  in 
Chapter  VIII,  we  have  RfsCt+Capa  where 
Ct-  Kdl(bf+aft)(p0+p0)+20tflPq] 
Ca=  nd[2(bf+aft)+2at£a 

For  symbols  see  discussion  of  floating  piston 
in  Chapter  VIII.    All  dimensions  may  be  expressed 
in  inches  and  pressures  in  Ibs.  per  sq.in.  in  place 
of  the  center  of  gravity  system  as  used  previously. 
The  resulting  friction  Rf  is  therefore  in  Ibs. 

The  drop  due  to  friction,  becomes, 

Rf 
pa  -  pa  »  -   where  Aa  is  the  area  of  the  floating 

Aa   piston  or  recuperator  cylinder  in  sq, 
inches  . 


576 


577 


GENERAL  THEORY  OF     Counter  recoil  is  divided  in- 
COUNTER  RECOIL.    to  two  periods,  (1)  the  first 

period  or  constant  orifice  period 
and  (2)  the  second  period  or 
buffer  period  wnere  the  main  re- 
tardation takes  place.   The  second  period  is  the 
critical  period  in  the  design  of  a  counter  recoil 
system,  since  with  field  carriages  the  stabilizing 
force  of  counter  recoil  is  relatively  small,  there- 
fore too  rapid  retardation  of  the  recoiling  mass 
will  cause  the  mount  to  be  unstable  on  counter  re- 

coil.  Let 

• 

A  =  effective  area  of  recoil  piston 

Ko=  contraction  factor  of  constant  orifice 

Kt*  contraction  factor  for  variable  orifice. 

WQ=  area  of  constant  orifice 

«x=  variable  area  of  buffer  throttling 
R  =  total  friction 

pa=  recuperator  oil  pressure 
Pa  *  Pa^  equivalent  recuperator  pressure  on 
recoil  piston 


C0*  •    =  throttling  drop  constant  for  con- 
176   stant  orifice. 

K*A* 
Cx=  -  *  throttling  drop  constant  for  buffer 

175   orifice. 

Ro=l?rsin0  +  R  a  resistance  constant. 

For  1st  Period  of  Counter  Recoil: 


Considering  the  notion  of  the  recoiling  mass, 
from  tne  initial  displacement  of  out  of  battery, 

we  have  dv 

pA  -Wrsin0-R=mrv  — 

but 

P  '• 

Assuming    the   throttling   drop   is   entirely 
through   the   constant   orifice  during   the   first   period 


578 


„«   .3      2 

KQA   V  dv 

of  recoil.        Hence   p^A  -  Wrsin  0-R=rorv  — 

175w*  dx 

or     p  v* 

• 

Since  pa  is  a  function  of  xtthe  equation  is 
not  possible  to  integrate  directly,  but  by  divid- 
ing the  constant  orifice  period  into  several  in- 
crements, and  taking  a  constant  air  pressure  equal 
to  the  mean  air  pressure  for  the  interval,  we  get 
a  very  close  approximation  of  the  true  velocity  of 
counter  recoil  by  the  following  solution,     9 

ror  »  dv 
dx  =  


Integrating,    we    have 

mrwo  CjjV*  Cov 


Substituting    for   the   base   10,    we  have 

Cov*  Crtv* 

logf(pa-R0) -]=logUpa-R0)' 

as 2  ivz  V>'4miuz 

"o  o     ^.omrwo 

From  this  equation,  knowing  the  velocity  at 
the  beginning  of  any  arbitrary  interval  and  with 
the  mean  recuperator  pressure  we  can  obtain  the 
velocity  at  the  end  of  the  interval.    It  will  be 
found  that  fairly  large  intervals  may  be  assured 
with  considerable  accuracy,  providing  the  air 
pressure  does  not  vary  greatly. 

The  velocity  curve  for  the  first  period  should 
be  continued  from  the  out  of  battery  position  to 
x  =  b-d,  where  d=  the  length  of  the  counter  re- 
coil corresponding  to  the  buffer  length. 

Let  vj,  =  velocity  of  counter  recoil  at  entrance 
to  buffer. 

For  2nd  Period  of  Counter  Recoil. 

The  recoil  displacement  is  d,  and  the  initial 


579 


velocity  v^ .    In  order  to  be  assured  that  the 
c 'recoil  is  completely  checked,  the  counter  re- 
coil energy  of  the  recoiling  mass  at  entrance  to 
buffer  should  be  dissipated  in  a  distance  somewhat 
less  than  d,  from  0.7  to  0.9d,  depending  upon  the 
design  constant  of  the  recoil  system  and  gun.   Let 
k  *  the  proportional  distance  of  d  that  the  recoil 
energy  is  to  be  dissipated  along. 

For  counter  recoil  stability  the  minimum  force 
during  the  buffer  period  is  obviously  obtained  by 
using  a  constant  force  during  the  entire  period. 

There  are  two  methods  consistent  with  counter 
recoil  stability: 

(1)  When  the  total  friction  is  small 
compared  with  the  overturning  force 
permissible  with  counter  recoil 
stability,  by  "bringing  the  recoiling 
mass  to  rest  into  battery  with 

t"he  friction  alone. 

(2)  Khen  the  total  friction  is  greater 
than  the  overturning  force  permissible 
with  counter  recoil  stability,  by 
bringing  the  recoiling  mass  to  rest 
into  battery  by  a  force  equal  to 

the  total  friction  minus  a  recoil 
pressure  exerted  on  the  recoil 
piston. 

In  method  (1)  obviously,  for  a  given  kinetic 
energy  of  the  recoiling  mass  at  entrance  into 
buffer,  the  recoil  displacement  during  the  buffer 
action  is  fixed. 
Method  (1) 

We  have  for  the  required  recoil  displacement 
during  the  buffer  action: 

1      2 

-m_vb 

where  k  =  0.7  to  0.9 


kR 

where  R  is  the  total  recoil  friction  (guide, 
stuffing  box  and  piston  friction). 


580 


The  length  of  the  buffer  in  the  recuperator 
cylinder  becomes, 

d'  -  —  d 
•I 

The  velocity  curve  is  evidently  a  parabolic 
curve  against  displacement,  that  is 

dv 

R  *  -  m_  v  — 
dx 

x  v 

/   Rdx  =  -  mr  /  v  dv 
b-d  vb 

R(x-b+d)  =  -£(v  -  v«) 


hence 

/     2R 
v  /  v*  --  (x-b+d) 

m 


Since  it  is  assumed  that  p  =  0,  we  have 


substituting  for  v,  we  have  wx  in  terms  of  the  dis- 
placement x  from  the  out  of  battery  position, 


vg  -  —  U-b+d) 
"x  •  M  "o   " 


--  (x-b+d)] 


where  b  =  length  of  recoil  in  ft.  and  d  = 
length  of  buffer  recoil  in  ft. 

x  -  counter  recoil  displacement  from  out  of 

battery  position  in  ft. 
The  throttling  drop 


581 


— —  through  the  constant  orifice  has  been  found 

1  7  i  M 

°   by  calculation  to  be  small  as  compared  with 
the  throttling  drop  due  to  the  buffer.   Therefore 
a  simplification  in  the  calculation  may  oe  made  by 
omitting  this  term,  hence 


.  —7-     and  substituting  for  v,  we  have 
13  •  2 


KtA    /mrvb-2R(x-b+a) 
HX  m  /  approx .  which  gives 

13.2     mrp^          the  required  throttling 
area  in  terms  of  the  displacement  of  counter  recoil, 
(x  is  measured  from  out  of  battery  position  in  ft; 
b  -  recoil  displacement  in  ft.  and  d  =  recoil 
buffer  displacement  in  ft). 
Method  2 

If  h  =  height  of  center  of  gravity  of  recoil- 
ing parts  above  ground  (practically  from  ground  to 
axis  of  bore)  and  wg  =  weight  of  entire  carriage 
and  gun  and  C's  -  distance  of  ws  from  wheel  contact, 
then  critical  stability  (at  0°  elevation)  we  have, 
(R-pA)h=Nsl^   using  a  factor  of  0.8,  we  have 

W  1 ' 
s  s 

R  -  p  A  *  0.8  where  R  is  the  total  friction, 

h 

but  now  a  function  of  the  re- 
coil pressure,  let  R  =  Ct+Cap  then 

Wsl' 

C  +P(C  -A)=0.8  — — 
1  h 

W  1 ' 
or    _  wsxs 

0.8  — C  h 

b      l 

p  =  where  C  -  guide  friction  as- 

°(Ct  ~  A)       sumed  independent  of  p  +  that 

part  of  packing  friction  in- 
dependent of  p. 

C  *  that  part  of  packing  friction  dependent  upon 
p.   The  counter  recoil  velocity  curve,  "becomes 
during  the  buffer  recoil, 


582 


v£  -  (x-b+d)   and  for  the  pres- 

n\r 

sure   p   in   the   recoil 


cylinder,    we    have 


K 


Pa  - 


175w« 


»  P 


hence 

KIA  v/"" 

1 

wx    -                / 

13.2 

K*A2v2 

(pa-p>- 


or  in  terms  of 
the  displace- 
ment x, 


(R-pA)  (x-b+d) 


2(R-p  A) 

175(pa-p)w*-K§A*  [vg (x-b+d)] 

mr 

Neglecting  the  constant  orifice  throttling  drop, 
we  have  the  following  approximate  formula 


12.2 


It  should  be  carefully  noted  that  if  v^ 
is  allowed  to  become  too  great  it  may  be  found 
very  difficult  to  prevent  the  final  check  of 
counter  recoil  without  shock  sure  with  even 
prompt  throttling  by  the  regulator  the  kinetic 
energy  ot  the  gun  may  overcome  the  the  opposing 
friction  and  cause  bumping. 


DESIGN  FORMULAE  ST.     In  the  preliminary  design 
CHAMOND  RECOIL  SYSTEM,  of  a  St.  Chamond  recoil  sys- 
tem, we  must  consider  the 
following. 
(1)     The  proper  weight  of  recoiling 

mass  wit"h  given  ballistics  and  allow- 
able recoil  at  maximum  elevation 
for  minimum  weight  of  the  total 
mount,  gun  and  carriage. 


583 


(2)  Tha  length  of  recoil  at  zero  ele- 
vation consistent  with  stability. 

(3)  The  total  resistance  to  recoil 
at  short  recoil  maximum  elevation 
and  at  long  recoil,  zero  elevation. 

(4)  An  estimation  of  the  guide  frict- 
ion and  packing  frictions  for  both 
recoil  and  counter  recoil. 

(5)  The  recuperator  reaction  re- 
quired to  hold  the  gun  in  battery  at 
Maximum  elevation. 

(6)  Limitations  of  recuperator 
dimensions . 

(7)  The  calculation  of  initial  air 
pressure  and  air  volume,  final  air 
pressure  and  air  volume.   From  this 
the  equivalent  air  column  length. 

(8)  The  calculation  of  strength  of 
cylinders  and  proper  thickness 
between  walls. 

(9)  The  layout  of  the  recuperator 
forging  distance  of  center  lines  of 
cylinders  with  respect  to  axis  of 
bore,  location  of  trunnions,  etc. 

(10)  The  calculation  for  maximum  and 
minimum  throttling  areas. 

(11)  The  calculation  for  entrance  chan- 
nel area  to  regulator  valve,  regulator 
dimensions  and  channel  areas  around 
and  leading  from  the  regulator  orifice. 

(12)  The  reactions  on  regulator  valve 
corresponding  to  deflections  at  maximum 
and  minimum  opening  and  the  design  of 
spiral  springs  and  belleville  washers. 

(13)  The  design  of  cam  mechanism  for 
changing  the  initial  opening  to 
regulator  valve  for  decreasing  the 
recoil  on  elevation. 

(14)  The  design  of  packing  for  float- 


584 


ing  piston,  recoil  piston  and 

stuffing  box. 

(15)  The  design  of  the  counter  re- 
coil and  chamber,  throttling  grooves 
and  constant  orifice  with  its  chan- 
nel leading  from  the  inside  end  of 
the  buffer  to  the  recoil  cylinder 

(16)  The  layout  of  gauge  and  pump 
details  and  all  other  details. 

(1 )  Proper  weight  of  recoiling  mass: 

From  "General  design  Limitations"  we  have 

»_  *  /  kk1  where 

"c 

k  T 

and 

.,  _  gUv  +  m  4700.)' 
2b 

m  =  mass  of  projectile 
I  *  mass  of  charge 
b  =  length  of  short  recoil  in  feet 
R  »  recoil  reaction  in  Ibs. 
g  =  acceleration  due  to  gravity  (ft/sec) 
• wc=  weight  of  carriage  excluding  recoiling 

mass  in  Ibs. 

k  may  be  obtained  from  table  in  Chapter  VII 
or  the  ratio 

wc 

—  may  be  computed  from  a  similar  well 

designed  type  of  carriage,  using 

a  somewhat  lower  value  of  "b"  according  to  the 
.judgment  of  the  designer  in  improving  the  weight 

efficiency  of  the  mount  proper  over  a  similar 
previous  design. 

(2)  Length  of  recoil  at  zero  elevation, 

Prom  pressure  curves  obtained  experimentally 
it  was  found  that  the  resistance  to  recoil  at  zero 
elevation  is  practically  constant  throughout  the 
recoil. 


585 


Let  b  =  length  of  horizontal  recoil  consistent  with 
stability  in  feet 

wv+w4700 
Vf  =  -  =  free  velocity  of  recoil. 

w  =  weight  of  projectile  in  Ibs. 
w  *  weight  of  charge  in  Ibs. 

Wf  =  weight  of  recoiling  parts  in  Ibs. 

v  =  muzzle  velocity  (ft/sec) 

Vr«  0.9  Vf(approx.)  =  velocity  of  restrained 

recoil. 

u  *  travel  of  projectile  up  bore  in  feet 
A  »  recoil  constrained  energy  =  7Jtirvr 
E  *  recoil  displacement  during  powder  period 

=  9  06. 

*  £.64 

"r 

Overturning  moment 
C  *  constant  of  stability  =  -  —  .  .  ,  . 

Stability  moment 

Wg  *  weight  of  total  mount,  gun  and  carriage 
lg  *  moment  arm  of  wg  about  spadepoint. 

d  =  perpendicular  distance  from  spade  point  to 
line  of  action  of  the  total  resistance  to 
recoil. 

Usually  £5  =  0,  cos  0  =  1  and  d  =  h  =  height  of 

axis  of  bore  above  ground. 

then 


Wrcos 


and  when  0  *  0,  we  have, 


1                       /               2AW_h 
b  -    E   +  —  [Wsls  ±  /Ofjjlp8 ] 

Tir  P 

•V  «»  *^ 

Ordinarily  the  constant  of  stability  will  be  as- 
sumed at  C  *  0.85. 

For  rough  estimates,  especially  wnere  the 
length  of  recoil  is  comparatively  long, 

Cs  Vrcos  0 


where  A 
2A 


586 


1  -7-     "  r  mrv! 

(3)  Resistance  to  recoil,  short  and  long  recoil, 

For  design  calculations,  Bethel's  formula  for 
the  total  resistance  to  recoil  is  sufficiently 
accurate.   Let  w  =  weight  of  projectile,  Ibs. 

Vfr  =  weight  of  recoiling  mass,  Ibs. 

M  =  travel  upbore  in  inches 

d  =  diam.  of  bore,  inches. 

v  =  muzzle  velocity  of  projectile  (ft/sec) 

Vj  =  max.  free  velocity  of  recoil  (ft/sec) 

bs  =  length  of  short  recoil  at  max.  elevation 
in  ft. 

bj,  =  length  of  long  recoil  at  zero  elevation 
in  ft. 

w  =  weight  of  powder  charge 
Now  for  the  free  velocity  recoil,  we  have 

w  4700  +  wv 
v   =  - 


then  at  maximum  elevation  and  short  recoil,  we 
have 


Ks'1.05[ 

8   u  +  (.  096*.  0003d  )m  — 
Ds  v 

(where  1.05  accounts  for  the  peak  effect  due  to 
throttling)  and  at  horizontal  elevation,  long  re- 
coil, we  have 


y —   the  peak  effect 
2g    hh+(.096  +.0003d)m-£   being  zero 

(4)     Guid*e  and  packing  frictions  -  Recoil  and 
Counter  Recoil:- 

In  the  calculation  of  guide  friction  during 
the  recoil  consideration  must  be  given  to  the  pinch- 
ing action  at  the  clips  due  to  the  pull  being 


587 


usually  below  the  center  of  gravity  of  recoiling 
parts.  Failure  to  consider  this  fact  will  give 
erroneous  results  for  the  friction  at  high 
elevation.   Also  the  recuperator  must  be  de- 
signed not  only  for  the  weight  component  at  max. 
elevation  but  the  friction  just  out  of  battery. 
Since  at  the  end  of  counter  recoil  we  have  the 
full  air  pressure  acting  on  the  recoil  piston, 
the  pinching  action  and  corresponding  guide  frict- 

ion being  a  factor  of  importance.  Let 

B  =  the  total  braking  including  the 

packing  friction  of  recoil  piston  and 
stuffing  "box,  Its. 
d^  =  distance  down  from  center  of  gravity 

of  recoiling  mass  of  line  of  action  of  B 
in  inches. 

pA  =  the  recoil  reaction,  Ibs. 
n  =  coefficient  of  guide  friction 
x^  and  xa  =  coordinates  in  direction  of  bore 
of  clip  reactions  measures  res- 
pectively from  center  of  gravity 
of  recoiling  parts  in  inches. 
1  =  x   +  x   =  distance  between  normal  clip 

reaction,  in  inches. 
_    Rfi  =  total  guide  friction. 

From  a  similar  previous  design,  a  value  of  b  and  1 
may  be  assumed. 

Prom  ChapterlV,  we  have 


2nBdb+nWrcos 


and  for  a  first 
approximation 


assume  x 


then 


R   = 


2n8ab 


but  K  =  B+Rg-  Wr  sin 
(K+Wrsin  0)1 


1+2  n 


588 


Knowing  K  and  assuming  1  and  b,  we  have 
BspA+Rs+Rp  where  Rg  -  stuffing  box  friction 

R  =  recoil  piston  friction. 

Since  the  design  and  estimation  of  packing 
friction  is  in  greater  part  based  on  previous 
empirical  data,  the  width  of  packing  and  cor- 
responding dimensions  being  entirely  an  empirical 
matter,  we  must  estimate  the  proper  value  from 
data  on  previous  satisfactory  packing  used  on  other 
guns. 

Now  in  general  the  packing  friction  may  al- 
ways be  represented  by  the  following  equationt- 
R  »  CY+C  p.   where  Ct  is  the  friction  component 
caused  by  the  springs  or  Belleville  washers.   Since 
the  object  of  the  Believilles  is  to  compensate  for 
the  deficiency  of  the  oil  or  air  pressure  normal 
to  the  cylinder  due  to  Poisson's  ratio,  if  we 
know  tne  maximum  pressure  and  assume  dimensions  for 
the  packing  we  may  compute  Ct  as  well  as  C?  and 
thus  estimate  the  friction  at  any  other  pressure. 

Maximum  pressures  normal  to  cylinder  should  be 


taken  as  follows: 

Pn 

k 

Hydraulic  piston 

°-88  Pmax. 

0.88 

Stuffing  box 

0-86  pmax. 

0.86 

Floating  piston: 

Air  side 

i-20  Pa  max. 

1.20 

Oil  side 

1-38  Pa  max. 

1.38 

Knowing  the  maximum  pressure  normal  to  the 
cylinder  "pn"  we  have, 


max. 


n  dC.05  b+.09(a  +  T^Pn  approx. 

0 


589 


s    s 


1         I 


tl 


590 


where  d  *  diam.  of  piston  rod  or  cylinder  in  inches, 
b  *  width  of  leather  contact  of  packing 
a  =  total  depth  of  one  silver  flange  of  pack- 

ing cup  in  inches. 
at  =  total  depth  of  outer  silver  flange. 

Prom  the  above  equation,  we  have 


Ci*c2  Pmax.  =  nd[.05  b+.09(a+ 


pn  where 


Pmax. 


tne   maxiraum   fluid    pressure,    Further, 


K   d   0.731 .05b  +  .  09  (a+  r^)  )proax  %  hence 

o 

a 


C  »  K  d[ .05b+.09(a+  — )](pn-  0.73  Pmax) 

a 

C  =  n  d  0.73[.05b+.09(a+  — )] 
8  2 

As  a  guide  for  suitable  values  of  a,  at  and  b 
with  given  maximum  fluid  pressures,  the  following 
table  has  been  constructed  of  values  used  in  cer- 
tain experimental  recuperators. 

75  m/m  Model  of  1916  MI. 


Pmax 

Ibs/sq. 

in. 


Recoil  piston 
Stuffing  box 
Floating  piston 


0.14"  0.18"  0.19"  5120 
0.14"  0.18"  0.19"  5120 
0.09"  0.18"  0.29"  1270 


3.3"   Model   of   1919. 


a     a 

b     Pmax  lbs/sq 

in. 

Recoil  piston    0.137" 

0.233"    5500 

Stuffing  box     0.137" 

0.233"    5500 

Floating  piston   0.137" 

0.194"    1850 

591 


4.7"  Model  of  1906 


Pmax 
Ibs/sq.in, 


Recoil  piston   0.128"  0.128"  0.335"  3820 

Stuffing  box    0.128"  0.128"  0.335"  3820 

Floating  piston  0.128"  0.200"  0.335"  1200 

4.7"  Model  of  1918. 

a      at      b  Pmax 

Ibs/sq.in, 

Recoil  piston   0.156"  0.218"  0.218"  4500 

Stuffing  box    0.156"  0.218"  0.218"  4500 

Floating  piston  0.125"  0.218"  0.35"  2300 


The  dimensions  a,  ai  and  especially  b  increase 
somewhat  with  the  diameters  of  the  cylinders  (that 
is  with  the  caliber  of  gun)  as  well  as  with  the  fluid 
pressure.   Let  C^  and  C^  be  the  packing   friction 
constants  for  the  stuffing  box. 

C"  and  C"  be  the  packing  friction  constants 
for  the  recoil  piston 

4-    U  n   to         D          ^R  ^°~     V  *•*  ^°O          /  \  V          •  W          /   P 

tnen  Kg   v  p        i        t  x      a 

=   Co      +Q     p 
i       °2P 

Therefore,  the  recoil  reaction,  becomes,  for  any 
pressure  pt 

(K+wrsin0)l 

pA  = (C0   +C  p) 

1+W  n  db 

If  we  assume  the  maximum  recoil  pressure  pmax 
corresponding  to  maximum  elevation  #roax  we  have 


C. 


d[.05b+.09(a+ 


where  pn  =  k  pmax,  k  being  obtained  from  the 
previous  table.    Therefore  the  effective  recoil 
piston  area,  becomes 


(K+»rsin0_ax)l  a 

k  n  d[-05 


592 


-   . 

(1*2  n  db)pmax 

In  general  pmax  =  4500  Ibs.  per  sq.in.  but  as 
we  shall  see  in  6,  with  guns  of  low  elevation  and 
with  reasonable  horizontal  stability,  the  max. 
pressure  may  be  necessarily  smaller  than  the  pack- 
ing limit  pressure  of  4500  Ibs.  per  sq.  inch.   The 
assumed  max.  pressure  for  calculation  of  packing 
friction  and  in  (5)  the  recuperator  reaction  is  at 
this  stage  a  questicn  of  experience. 

The  guide  friction  when  the  gun  is  in  battery 


becomes,       2n  Brdb 


R  = 

l+2nr 

where  Bv  =  paA-(R-s  +  Rp)  i.e.  the  tension  of  the  rod 

in  battery 

r  =  distance  down  to  mean  friction  line 
1  =  distance  between  clip  reaction  in  battery. 
n  =  coefficient  of  friction  =  0.15  to  0.2 
b  =  distance  from  center  of  gravity  of  re- 

coiling parts  to  line  of  action  of  By 
Further  the  value  of  Rg^R   in  battery  becomes, 


To  compute  the  drop  of  pressure  across  the  float- 
ing piston  friction,  we  have 

Rf *+C"'+C£ '  — — —    approx. 
while  pa~Pa  *  — 
hence     2C"'  +  (C 

Pa  =  — h — : 


,  _  (2Aa-Cg)pa-2Ct 
C  +2A 


. 


which  gives  the  air  pressure  in  terms  of  the  re- 
cuperator pressure  (for  recoil  computations)  or 


593 


the  recuperator  pressure  in  terms  of  the  air  pres- 
sure in  terms  of  the  air  pressure  (for  counter  re- 
coil computations). 

In  a  preliminary  layout,  we  are  not  greatly  • 
in  error  in  assuming  either 


Pa-Pa  = 


approx.  drop  due 
or 


to  floating  piston. 


"a 

(5)   The  recuperator  reaction  required  to  hold  the 
gun  in  battery 

To  ensure  a  sufficient  margin  for  the  holding 
of  the  gun  in  battery  and  overcoming  the  friction 
in  battery,  an  excess  of  20*  to  30*  is  used  over 
the  minimum  recoil  reaction,  hence 


2nKydb 

_       Rg   =  -  ;    k  =    1.2   to   1.3,n»0.15 
1+2   nr 

Kv 
and     B8+Rp=C,+  C,   — 

where   Ct=0.  15*  (dr+d)  (dp+d)[.05b  +  .09(a+  T 

a 


C  =0.73n(dr+d)l  .05b 

t 

and  for  a  trial  value,  Pmax  =  4500  Ibs.  per  sq.in 

ps 

A  =  - 
4500 

d  *  diam.  of  recoil  piston 

dr=diam.  of  piston  rod 

dfc38  distance   down   from  center   of  gravity   of   re- 

coiling parts   to  line  of   action  of   Kv 
Substituting    in    (1),    we   have, 

2nk   Kvdb  ITV 

~  -   +Ct+C2  ~ 
1+2   nr  A 


594 


kWrsmn0_ax+C 

bence  K  =  -  -  -  Z£*  —  i  -  (2) 

2nk  d    C 


l+2nr    A 

(6)   Limitations  of  Recuperator  Dimensions: 

In  the  design  of  a  recuperator  layout,  we 
must  consider  the  proper  ratio  of  area  of  re- 
cuperator cylinder  to  effective  area  of  piston, 
the  limitations  of  areas  based  on  this  ratio  and 
on  the  maximum  allowable  packing  pressure  in  the 
recoil  cylinder  as  well  as  on  the  difference 
between  the  horizontal  pull  and  recuperator  re- 
action at  maximum  elevation.   If  now, 

wn=  max.  area  of  orifice  at  horizontal  re- 
coil in  sq  .  in. 

A  =  effective  area  of  recoil  piston  in  sq.in. 

Aa  *  area  of  recuperator  cylinder  or  floating 
piston  in  sq.  in. 

V  =  max.  recoil  velocity  in  ft.  per  sec. 

Pn=  horizontal  pull  in  Ibs. 

ps=  pull  at  maximum  elevation  in  Ibs. 

Kv=  recuperator  reaction  at  maximum  elevation 
in  Ibs. 

1 

K  =  -  =  reciprocal  of  orifice  contraction 
0.773   constant. 

wc=  channel  or  port  area  at  cross  section  beyond 

regulator  valve  in  recuperator  cylinder 
in  sq.  in. 

Ws*  area  of  channel  section  through  diameter 

of  regulator  valve  in  sq.  in. 
a  =  entrance  area  to  regulator  valve. 
d  =  diam.  of  regulator  valve. 

r  =   ratio 

Floating    Piston   Area 


of 


Effective  Area  of  Recoil  Piston 


Now  the  maximum  throttling  area  w^  i»  the 
limiting  throttling  area,  since  all  constant  port 


595 


or  channel  areas  in  the  recuperator  should  bear  given 
ratios  with  respect  to  this  area  and  must  be  suf- 
ficiently large  as  compared  with  wh  so  that  there 
is  no  appreciable  throttling  through  them,  or  loss 
of  head  due  to  friction  or  acceleration. 

The  following  table  gives  ratios  of  channel 
or  port  areas  in  the  recuperator  with  respect  to 
the  .maximum  throttling  area  w^  and  the  area  of 
the  recuperator  cylinder  or  floating  piston. 


Model 

"h 

"c 

we 
C     =»     
1           a, 

wh 

*c 

C*T 

*a 

-. 
*>*-. 

4.  V  "-M.  19O6 

0.  38O 

1.61 

4.  235 

0.  126 

0.  608 

4.7"M.  1918 

0.  854 

3.6-7 

4.  3OO 

O.  207 

O.  64O 

3.  3"-M.  1918 

0.267 

1.  11 

4.  160 

0.  186 

0.  470 

75»»-M.  1916 

0.  1*75 

o.  7s 

4.  47O 

O.  120 

0.371 

4.  3OO 

o.  160 

From  the  above  table,  the  following  design 
constants  will  be  used  based  on  satisfactory  layouts: 


C  =  —  =  4.3 

1  wh 
Now 


=  f  =  0.16 
C. 


c   C.       , 

*•'  a  •  -0373  r " 


Considering  the  throttling  through  the  regulator 
orifice,  we  have 


2 

»h 


KVv* 


(1) 


175(pn-Kv) 

and  substituting  for  wh=.0373  r  A,  we  find 
4.11  AV2 


that  is  r  =  2.62  V 


Pb'Ki 


(2) 


Hence,  the  ratio  of  the  recuperator  area  to  ef- 
fective area  of  recoil  piston  varies  as  the  square 
root  of  the  effective  area  of  the  recoil  piston, 


596 


and  for  minimum  recuperator  area  we  must  have  min- 


imum effective  area  of  recoil  piston. 


Hence  the  recuperator  area  always  varies  as  the  - 
power  of  the  effective  area  of  the  recoil 

piston. 

Now  for  minimum  neight  of  the  recuperator 
forging  it  is  important  that  the  cylinder  areas 
be  made  as  small  as  possible.   Therefore  the  re- 
coil piston  area  in  general  is  limited  by  the 
maximum  allowable  pressure  in  the  recoil  cylinder 
consistent  with  the  packing  pressure  limitations. 

If  the  packing  limiting  pressure  is  taken  at 
4500  Ibs.  per  sq.  in.,  then 

A  =  -£i- 

4500 
Substituting   for   A,   we   have 

.039V  /  -  ^« 
Pb~Kv 

How  r  is  limited  by  the  following  considerations: 
(a)     In  an  ordinary  layout,  the 
initial  volume  of  the  air,  may 
be  represented  as  the  sum  of 
_,     air  column  in  the  recuperator 

cylinder  plus  the  air  column  in 
the  air  cylinder,  that  is 


where  kt  =  initial  air  column 
length  in  re- 


cuperator length  of 

recoil 
and  k  ~  air  column  length  in  air 

cylinder  length  of  recoil 
k  may  be  obtained  from  the  fol- 


597 


lowing  table: 


b 

X  1 

b 

b'..x 

0  »«x 

b  ' 

k»      b 

wax 

4.  7"-M.  1918 

40 

28.  27 

1.  41§ 

o.  7o;; 

4.  7"-M.  1906 

7o 

56.50 

1.  240 

0.  807 

75»m-¥.  1916 

46 

43*08 

1.  07O 

0.937 

3.  3"-M.  1918 

60 

47.67 

1.260 

o.795 

1.  246 

0.811 

Therefore  «e  may  assume  Ki=0.8  and  kf  will  be 
assumed  ka=  0.7,  hence  kt  +  ks  =  1.5 

The  initial  volume  as  shown  in  (7)  may  be 
represented  in  the  recoil  piston  displacement  and 

the  ratio  of  the  final  and  initial  air  pressures. 

Paf 
If  o  =  and  k  =  (1  to  1.41  use  1.3) 


then 


when 


m  =  1.5 


(6) 


=  3.73 


.k  -  1 


2.0 


3.0 


=  2.42 


-  1 


1.75 


-  1 


Using  a  ratio  of  m  =  1.5,  and  combining  Eq.  4 
and  5,  we  have  3.73  Ab=1.5Aab  hence 


=  r  =  2.5 


598 


Paf 
If  a  lower  value  of  m  =  -  is  used,  then  r  >  2.5 

Pai 
where  as  with  a  higher  ratio,  r  <  2.5 

If  we  continue  increasing  m  for  the  minimum 
permissible  value  of  r  we  soon  arrive  at  the 
limitation  where  the  maximum  possible  recoil  of 

the  floating  piston  limits  the  ratio  r.  since  Aa 

amin 

bmax  -  A  b-   Then       amin     b 


max 

On  the  other  hand  to  obtain  a  value  of  rmjn= 
1.25,  would  require  a  high  value  of  m,  approx.  a 
3.0,  and  the  temperature  rise  of  the  air  would  be 
excessive  and  very  injurious  to  the  floating  pis- 
ton packing. 

Further,  at  horizontal  recoil,  where  a  stability 
slope  for  the  total  resistance  to  recoil  is  highly 
desirable,  we  have  the  minimum  throttling  drop 
due  the  small  value  of  the  pressure  in  the  recoil 
cylinder  at  horizontal  recoil.   The  peak  effect  of 
the  throttling  drop  is  thereby  reduced,  and  since 
the  pressure  in  the  recoil  cylinder  is  the 
throttling  drop  plus  the  recuperator  pressure,  a 
large  increase  in  the  final  air  pressure  over  the 
initial  will  overbalance  the  decrease  in  the 

throttling  drop  towards  the  end  of  recoil.  There- 

fore, a  large  value  of  m  =  ¥-&£  must  result 

in  a  rise  of  the  total     pai 

resistance  to  recoil  towards  the  end  of  recoil 

which  is  entirely  inconsistent  with   the  re- 

quirements for  horizontal  stability 

If,  therefore,  at  horizontal  recoil,  we 


limit  the  ratio  m  =s   to  that  value  which  gives 

us  a  constant 

resistance  throughout  the  recoil,  we  have 


neglecting  the  slight  throttling  drop  at  the  end 


599 


of  horizontal  recoil.   Substituting  for 
wh».0373  r  A         K 


0.773 


and 


we  have,  2 

0.145(m-l)r2=6.87  -      (8) 
Pai 

From  the  following  table,  knowing  V  and 
pa^  and  from  the  above  equation  (m-l)ra,  we  can 
readily  obtain  r  or  corresponding  value  of  m. 


3  CYLINDERS 


2  CYLINDERS 


r 

r2 

r2 

6.  88 

(m-1) 

V2 
Pai 

m-1 

V* 
*ai 

1.5 

2.  "75 

.3273 

1.  147 

.3787 

1.547 

.5063 

1.6 

2.  §6 

.3*724 

1.015 

.3780 

1.  345 

.  5001 

1.7 

2.  89 

.  4204 

.  910 

.  3824 

1.  188 

.4994 

1.  8 

3.24 

.  4714 

.  824 

.  3884 

1.064 

.  5018 

1.9 

3.61 

.5252 

.  752 

.3950 

.  960 

.  5041 

2.0 

4.  OO 

.5819 

.  694 

.  4038 

.  880 

.  5120 

2.  1 

4.  41 

.  6416 

.  645 

.  4138 

.  810 

.5197 

2.2 

4.  84 

.7041 

.599 

.  4218 

.750 

.5280 

2.3 

5.  29 

.7696 

.  560 

.  4310 

.  7oo 

.5287 

2.4 

5.16 

.8380 

.527 

.  4416 

.653 

.54*72 

2.5 

6.  25 

.  9093 

.497 

.4519 

.612 

.5565 

As  a  check,  we  may  obtain  the  ratio  m  con- 
sistent with  horizontal  stability  from  another 
point  of  view. 

At  the  end  of  recoil,  we  have,  neglecting 
a  small  throttling  drop,  Paf*=Ph  wnereas  the 
initial  reaction,  we  have,    AsK   hence 


Paf   Ph 
m  =  -  =  —   which  gives  immediately  the 

a*    v    maximum  ratio  tor  m  or  the 
minimum  ratio  for  r,  consistent  with  the  require- 


600 


nents  for  horizontal  stability.     „ 

Again  the  maximum  value  of  m  -  depends 

n   • 

further  upon  the  heating 

limitation  (i.  e.  the  permissible  rise  of  tem- 
perature of  the  air  in  the  recuperator  forging). 
Since  the  question  of  heating  depends  upon  the 
various  factors,  as  radiation  through  the  cylinder 
walls,  the  frequency  of  firing,  etc.  we  must  as- 
sume by  experience  for  the  given  type  of  gun,  the 
maximum  allowable  temperature  use  during  the  com- 
pression of  the  recuperator  air.    Thus,  using 
a  ratio  m  =  2,  we  have 

k-l 

T  Paf   k     0.23 

—  ( )    =  2      Assuming  a  mean  temperature 

Tm  Pai  at  25°  Centigrade,  we  find 

T  -  298  *  2°'*8  =  349°,(k=1.3)   therefore,  the 
rise  of  temperature  during  a  recoil  stroke  becomes, 
T  -Tn=51°C  or  92°F.   This  rise  of  temperature  is 
not  excessive.   If  the  rapidity  of  fire  is  great 
the  mean  temperature  will  rise.   The  quantity  of 
heat  generated  is  the  work  done  on  the  air 
divided  by  the  mechanical  equivalent  of  heat. 
Now  if  the  mean  temperature  has  risen  to  its 
constant  maximum  value,  then  the  heat  generated 
during  the  firing  stroke,  must  be  dissipated 
by  radiation  through  the  cylinder  walls  during 
the  period  of  loading,  the  temperature  gradient 
varying,  decreasing  during  the  process  of 
radiation  through  the  cylinder  walls. 

Thus  we  see  we  have  two  aspects  for  the 
maximum  ratio  of  m  and  the  corresponding  minimum 
value  of  r: 

(1)     To  maintain  at  best,  a  constant 
resistance  to  recoil  throughout  the 
recoil  at  horizontal  elevation, 
rather  than  a  rise  in  the  over- 
turning force  at  the  end  of  recoil 
which  would  be  entirely  opposite 


601 


to  the  requirements  for  the  proper  stability 
slope  at  horizontal  recoil.   This  limitation 
for  r  is  of  special  importance  when  pn~Ky  is 
small,  we  have  rfflin  determined  by  the  equation, 

(m-l)r  =  6.87  where  we  may  obtain  •  d  r 

ai  for  various  values  of  (m-l)r* 
from  the  previous  table. 

(2)     On  the  other  hand,  when  Ph~Kv 

is  large  as  with  guns  of  low  elevat- 
ion and  where  we  have  a  good  margin 
of  stability,  the  peak  effect  in 
the  throttling  becomes  larger  or 
the  ratio 

Pv> 
— IL  is  larger,  therefore,  a  higher 

"v  value  of  m  can  be  used.   In  such 
a  case  we  become  limited  by  the 
rise  in  temperature  of  the  air 
during  the  firing. 

(a)  When  (2)  becomes  the 
limitation  we  may  use  a  higher 
value  of  m  and  therefore  a 
lower  value  of  r. 

(b)  The  expansion  of  the  oil 
varies  considerably  with 
temperature,  different 
viscosities,  the  rapidity  or 
frequency  and  continuity  of 
fire,  etc.  Therefore  the 
floating  piston  will  have 
various  initial  displacements, 
resulting  in  different  initial  air 
pressures  and  most  important 

in  unsatisfactory  functioning 
of  the  buffer  on  counter  re- 
coil, the  buffer  action  start- 
ing at  various  different  points 


602 


on  counter  recoil,  unless  the 
ratio  is  made  sufficiently  large, 
since  with  a  large  bulk  of  oil, 
temperature  changes  and  consequent 
expansion  is  less. 

(c)     Due  to  the  wear  of  the  pack- 
ing on  the  floating  piston  it 
has  been  customary  to  limit  the 
maximum  velocity  of  the  float- 
ing piston  to  not  over  12  ft. 
per  sec.  though  it  is  believed 
that  the  packing  may  be  designed 
to  withstand  a  surface  velocity 
of  20  ft.  per  sec. 
Therefore,  in  conclusion,  from  a  consideration 

of  (a),  (b)  and  (c)  rmin  ordinarily  should  be 

limited  to:  rmin=1.3  to  2.0 

In  difficult  designs,  however,  the  proper 

minimum  value  of  r  should  be  determined  more  from 

a  consideration  of  aspects  (1)  and  (2)  in  (a)  rather 

than  (b )  and  (c). 

Limitations  for  the  maximum  value  of  r: 

For  very  large  values  of  r,  slight  changes 
in  the  quantity  of  oil  due  to  leakage,  will  have 
a  marked  effect  on  the  relative  initial  positions 
of  the  floating  piston  and  recoil  piston.   Further 
it  would  be  difficult  to  gauge  slight  variations 
in  the  quantity  of  oil  due  to  the  relatively  small 
motion  of  tbe  floating  piston  which  moves  the 
gauge  rod . 

But  most  important  from  a  point  of  economy 
in  the  weight  of  recuperator  forgings  and  very 
often  in  a  satisfactory  layout,  a  too  large  value 
of  r  becomes  prohibitive. 

fle  limit  the  max.  value  of  r  to,  rmax=3.5 
Rence  the  design  limitations  for  r  becomes  for 
an  ordinary  layout  with  3  cylinders  r  =>1.8  and 
r-^3.5.    When  only  two  cylinders  are  used  we  have 


603 


tbe  following   considerations: 

(1)     If  the  length  of  the  recuperator 
air  cylinder  has  the  same  total 
length  as  the  recoil  cylinder,  we 
x       have  ki=0.8  and  ka*0,  hence 
k 


A=0.8  A   hence  r  = 

0.8  i 


Ab=0.8  A  b  hence  r 


. 
mk  -  1  mk  -  1 

If  m  =  2  a  heating  limit  on  the  ratio  of  com- 
pression, we  have    1 

k 

—  -  -  =  2.42   for  k  =  1.3 

I 

m*  -  1 

and 

2.42 

r  «  -  =  3.025 
0.8 

On  the  other  hand  (in  consideration  of  a 
constant  recoil  reaction  for  stability  at 
horizontal  elevation)  if  we  decrease  m  to  m  *  1.3, 
we  have     3.73 

r  *    •  -  4.67  which  gives  a  very 
0  •  o 

bulky  recuperator  with 

too  snail  relative  displacement  of  the  floating 
piston  as  compared  with  the  recoil  displacement. 
Therefore,  if  only  two  cylinders  are  to  be 
used  and  of  tbe  same  length  we  are  peculiarly 
limited  by  bulk  and  a  small  movement  of  the 
floating  piston  as  compared  with  the  recoil  piston, 
on  the  one  hand,  while  with  a  decrease  in  the 
ration  r,  the  final  air  pressure  is  increased  and 
overbalances  the  peak  of  the  throttling  plus  the 
initial  air,  thereby  giving  a  rise  in  the  recoil 
reaction  towards  the  end  of  recoil  at  horizontal 
elevation. 

Therefore  as  a  compromise,  if  two  cylinders 
must  be  used  of  the  same  length  we  may  take 
r  =  3.5  and  m  =  1.8  and  r  is  to  be  considered 


604 


constant  for  this  combination.    Thus  we  see  by 
the  use  of  two  cylinders  only  and  of  the  same 
length,  the  ratio  cylinders  become  excessive 
if  aoderate  compression  ratios  are  maintained, 
whereas  for  moderate  cylinder  ratios  we  must 
maintain  high  compression  ratios  which  cause 
undue  heating  and  a  rise  of  the  recoil  reaction 
at  horizontal  elevation. 

A  more  satisfactory  combination  for  two 
cylinders  only,  is  by  use  of  longer  recuperator 
cylinder  than  recoil  cylinder.   This  is  usually 
feasible  especially  for  guns,  where  the  tube  is 
long.   A  satisfactory  approach  to  three  cylinders 
may  be  had  by  the  use  of  a  sufficient  overhang 
of  the  recuperator  cylinder. 

The  air  column  in  the  overhang  can  be 
reasonably  assumed  at  0.5  the  horizontal  recoil. 
Therefore  in  the  equation  VQ=Aa(kt+k2.)b .  We  may 
assume  as  before  kt=0.8 

k  »0.5 

2 

=  1.3 
hence  kt  +ks 

Hence  with  a  ratio  of  pressure  expansion  in  the 
recuperator  m  =  1.5,  we  have  3.73  Ab=1.3Aab  hence 

Aa 

—  =  r  =  2.87.   By  increasing  the  ratio  of  ex- 
pansion we  may  limit  the  minimum 
r  to:  *n)in=2'5.   From  the  above  it  is  evident 
that  though  it  is  not  feasible  to  use  a  recuperator  air 
cylinder  as  long  as  the  combined  length  of  a 
separate  recuperator  and  air  cylinder,  on  the  other 
hand  the  minimum  ratio  of  r  is  greater  than  with 
an  ordinary  layout  and  with  only  two  cylinders, 
the  total  weight  of  t"he  forging  may  exceed  that 
of  these  cylinders.    Hence  if  two  cylinders 
are  to  be  tried  in  place  of  three,  preliminary 
separate  layouts  for  the  two  combinations  should 
be  worked  out  in  view  of  minimum  weight  and 


605 


satisfactory  layout  before  either  plan  is  adopted. 
Therefore,  in  a  design  layout  we  start  with 

which  determines  the 

Ph~kv 
'a 

ratio  —  providing  it  falls  within  the  limits 
A 

rmin  and  rmax*   Hence  the  recoil  area, 
becomes, 

A  =  — — 
4600 

Anti-aircraft  Guns; 

Anti-aircraft  guns  are  the  most  difficult 
to  design  without  having  excessive  bulky  re- 
cuperator forgings,  Pn-Kv  becomes  small,  since 
Ky  is  larger  to  hold  the  gun  in  battery  at  max- 
imum elevation  and  p^  is  small  to  satisfy  stability 

at  zero  elevation,  further  ps  is  large,  there- 
fore r  is  in  general  large,  usually  3  or  above. 

If  r  exceeds  3.5  using  3  cylinders,  we  must 
increase  Pt]~Kv  either  by  reducing  Ky  and  then  al- 
lowing the  gun  to  return  slower  into  battery  at 
maximum  elevation  and  with  a  smaller  margin  of 
excess  battery  reaction  of  the  recuperator  at 
maximum  elevation  or  preferably  increase  pn  at 
the  slight  sacrifice  of  stability  at  zero 
elevation,  -  in  this  case  we  have 

Ph=Kv+. 000912V8  — — 
3.52 

It  will  be  rarely  found,  however  that  r  ex- 
ceeds 3.5. 
Howitzers: 

With  Howitzers,  we  again  meet  the  condition 
of  a  large  Kv  but  since  horizontal  stability  is 
not  a  consideration,  pn  may  be  relatively  large, 
and  therefore  pn~Kv  still  may  be  maintained 
large. 


606 


r  is  generally  medium  or  small  at  the  sacri- 
fice of  horizontal  stability. 

Guns : 

With  guns,  r  is  in  general  small,  usually 
from  2  to  3.   If  r  decreases  below,  2.0  to  1.8 
using  3  cylinders,  or  2.5  using  2  cylinders  with 
a  longer  recuperator  cylinder  than  tfce  recoil, 
the  effective  area  of  the  recoil  piston  is  now 
determined  by  the  formula: 
x 

A  =  0.2425  — (pu  -  Kw)      and  the  maximum  pres- 

v* 

_   sure  in  the  recoil 
P  g 

cylinder,  becomes,  pmax=  — 

A 

Howitzers  and  Guns  on  Same  Carriage: 

When  Howitzers  and  guns  are  adopted  for  t"he 
same  carriage, Ky  must  be  large  for  the  howitzer 
and  ph  small  for  the  horizontal  stability  of  the 
gun.  Therefore  pn~~Kv  is  small,  ps  large  and  we 

meet  exactly  similar  conditions  as  with  anti- 
aircraft guns. 

Hence,  for  this  combination,  r  is  in  general 
large,  usually  3  or  above.    If  r  exceeds  3.5,pj, 
must  be  increased  or  Kv  decreased,  these  values 
being  connected  by  the  relation 


KV+.000912V« 


(7)   Calculation  for  air  pressures  and  volumes: 

From  (5)  we  have  for  the  initial  recuperator 
reaction 


Kv  - 

k 

Wrsin 

^m  a 

X+Ci 

where  k 

Pmax  may 
at  4500 

in.   a 
+.09(a+  s±) 

B 

=  1.2  to  1.3 
be  assumed 
Ibs.  per  sq. 

JPmax 

1 

C 

2nkK 

vdb 

cz 

A 
+d)[.05b 

^O.lSn 

nr 
WP 

C  =0.73n(d_+d)[.05b  +  .09(a  +  —)] 

2 

PS 

A  =  =  effective  area  of  recoil  piston 

4500 

a,  at  and  b  are  contact  lengths  of  the  pack- 
ing in  inches. 

d^-  distance  down  from  center  of  gravity  of 
recoiling  parts  to  line  of  action  of  Kv 
in  inches. 

1  =  distance  between  clip  reactions  in  inches 
n  =  0.15 

r  =  distance  down  from  center  of  gravity  of 
recoiling  parts  to  mean  friction  line 
in  inches. 

For  guns  of  low  elevation  and  reasonable  stability 
where  r  falls  below,  r    we  have 


A  =    0.2425  — (p     -   Ky)        and 

V 

Id  =  /0.7854A+dr 

2 

r 
A  =  0.2425  — (ph-Kv)    and  the  maximum  pressure 

^*  in  the  recoil  cylinder 

is  thereby  reduced  to: 

PS 
Pmax  =  —*  and  the  area  of 

A 

the  recuperator 

cylinder  becomes,  Aa=rminA.  With  an  ordinary 
layout  using  3  cylinders,  rmin=2  and  with  2 
cylinders  and  an  overhang,  rmjn=2.5.   Knowing  Aa, 
we  have  for  the  inside  diameter  of  the  re- 
cuperator, 

Da  : 

0.7854 

To  determine  the  diameter  of  recoil  cylinder, 
we  must  know  the  area  of  the  piston  rod,Ar.   Then 

/A+Ar 

D  *   /• where   A_   is   determined   as   follows: 

0.7854  r 


608 


If  6  is  the  total  hydraulic  braking  including 
the  joint  frictions  at  the  stuffing  box  and  re- 
coil piston,  we  have  B+Rg=K+Wrsin#  where 


2ndb 

)=K+Wrsin 

l+2nr 


The  maximum  stress  in  the  rod  is  at  a  section 
at  the  lug,  at  maximum  acceleration  of  the  recoil- 
ing parts  and  at  maximum  elevation.  If 

T   =  the  tension  in  the  rod  at  the  lug 
L 


K+Wrsin0ffiax 

2ndb 
1+  


(1-  =-)+  Pb  r- 


l+2nr 

where  We  »  weight  of  rod  and  recoil  piston 

pb  =  total  maximum  powder  pressure  on  base 

of  breech. 

If  fDax  is  the  maximum  allowable  working  fibre 
stress  in  the  rod,  we  have       f 

*  max 

and  Aa  *  rA.    Now  with  guns  of  low  elevation, 
Ky  is  snail  and  ph  relatively  large,  hence  the 
difference  pn~Kv  is  large  and  ps  is  small.  There- 
fore r  becomes  small. 

In  (7)  tables  and  a  chart  has  been  con- 
structed giving  values  of  m  and  r  for  different 
air  column  lengths,  the  air  columns  being  expressed 


609 


in  terns  of  a  ratio  of  the  length  of  air  column 
divided  by  the  length  of  recoil,  that  is 

j  «  -  where  1  =  length  of  ai*  column, 

b 

b  =  length  of  recoil. 

A  maximum  limitation  of  m,  based  on  a  moderate 
temperature  rise  of  the  air  during  the  recoil  and 
a  constant  reaction  throughout  the  recoil  at 
horizontal  elevation  (i.  e.  no  increase  of  the 
recoil  reaction  at  the  end  of  recoil,  )  will  be 
taken  at  1.8.    Evidently  for  different  air  column 
lengths,  we  will  have  different  minimum  values 
of  r,  corresponding  to  a  waximum  value  of  m  = 
1.8. 

The  longer  the  air  column  lengths,  the  lower 
the  ratio  r. 

If  r  falls  below  the  minimum  allowable  value 
of  r  (i.e.  the  r  corresponding  to  m  =  1.8)  for  a 
given  air  column  length,  r  becomes  a  constant, 
and  the  area  of  the  recoil  piston  must  be  in- 
creased according  to  the  formula*-  then 

Kv 
pa^  =  -j-  Ibs.  per  sq.in.  initial  recuperator  pres- 

sure intensity. 
Hence  the  initial  air  pressure  becomes, 


Pai  s  Pai  +  '  a*d  Paf 

Aa 


t 
C||l  =  1.46nda[.05b  +  .09(a  +  y)] 

Paf 

Next  to  determine  the  proper  ratio  of  m  =  - 

Pai 
we  must  consider  the  following: 

The  initial  volume  Vo  is  expressed  by  either 
memoer  of  the  equation, 


610 


»  Agl  where  k  »  1.3, 


b  =  length  of  recoil 
1  »  length  of  air  column, 
reduced  to  an 
equivalent  cross 
section  A. 


Since  r  *  — ,  we  have 
A 


r  -  •  rj,  where  j 
b 


The  following  tables  give  the  relation  of 
o  and  r  for  various  values  of  j. 


Pax 
Pai 


0.8  r 


1.3 


.77 


m 

r 

1.  2 

9.575 

1.4 

5.476 

1.6 

4.  116 

1.8 

3.  442 

2.0 

3.002 

2.  2 

2.746 

2.  4 

2.  546 

2.6 

2.  401 

2.  8 

2.  282 

3.0 

2.  188 

3.2 

2.  Ill 

3.4 

2.047 

3.5 

2.018 

611 


Values   of   m  and   r   for  j    =  1.1 


1.  lr 

1.  lr 

1.  lr 

r 

1.  lr 

1.  lr 

• 

log... 

Klog  

m 

-1 

1.  lr-1 

1.  lr-1 

1.  lr-1 

1.  8 

1.  98 

.98 

2.020 

.  30535 

.39696       . 

2.  494 

2. 

2.  2 

1.  2 

1.833 

.  26316 

.34211 

2.  198 

2.  2 

2.42 

1.  42 

1.  704 

.23147 

.  30091 

1.999 

2.4 

2.64 

1.  64 

1.  61 

.20603 

.  26888 

1.857 

2.6 

2.86 

1.86 

1.54 

.  18686 

.  24291 

1.750 

2.  8 

3.  08 

2.  08 

1.  48 

.  17056 

.  22173 

1.666 

3.2 

3-52 

2.  §2 

1.  4 

.  14520 

.18876 

1.  544 

3.4 

3-  "74 

2.  74 

1.36 

.13513 

.  17567 

1.499 

3.5 

3.85 

2.  85 

1.35 

.  13066 

.  16986 

1.4^9 

Values   of   m   and   r   for   j 

»    1.3 

1.3' 

1.  3r 

1.3r 

r 

1.3' 

1.  3r-i 

Klog  

• 

1.3'-1 

1.3*-1 

1.  3r-l 

1.  8 

2.34 

1.34 

1.  746 

.  24204 

.31465 

2.064 

2.0 

2.  60 

1.  60 

1.  625 

.  21085 

.27411 

1.  880 

2.  2 

2.86 

1.86 

1.  538 

.  18696 

.24305 

1.  850 

2.  4 

3.  12 

2.  12 

1.  472 

.16791 

.  21828 

1.653 

2.6 

3.38 

2.  38 

1.  420 

.  15229 

.  19798 

1.577 

2.  8 

3.64 

2.64 

1.379 

.  13956 

.  18143 

1.519 

3.0 

3.90 

2.  90 

1.345 

.  12972 

.16734 

1.470 

3-2 

4.16 

3.16 

1.  316 

.  11926 

.  15504 

1.429 

3.  4 

4.  42 

3.42 

1.  292 

.  11126 

.  14464 

1.395 

3.5 

4.55 

3-55 

1.  282 

.  10789 

.  14026 

1.  381 

-sol   *i   fcfl 


612 


Values  of  n  and  r  for  j  =  1.5 


l.Sr 

l.Sr 

l.Sr 

r 

l.Sr 

l.Sr  1 

Klog  —  —  - 

m 

1.  8 

2.  70 

i.7o 

1.  588 

.  20085 

.  26111 

1.  824 

2.  0 

3.  oo 

2.  00 

1.  50O 

.17609 

.  22892 

1.  694 

2.  2 

3.  30 

2.  30 

1.435 

.  15685 

.  20391 

1.  599 

2.  4 

3.  60 

2.  60 

1.385 

.  14145 

.  18389 

1.527 

2.6 

3-  90 

1.345 

1.345 

.  12872 

.16734 

1.  470 

2.  8 

4.  2O 

3.  20 

1.313 

.  11826 

.  15374 

1.  425 

3.0 

4.  50 

3.50 

1.  286 

.  10  9  2  4 

.  14201 

1.387 

3.2 

4.  8O 

3.  80 

1.263 

.  10140 

.  13182 

1.355 

3.4 

5.  10 

4.  10 

1.  244 

.  09482 

.  12327 

1.  328 

3.  5 

5.  25 

4.  25 

1.235 

.09167 

.  H917 

1.  316 

Values 

of  m  and  r  for  j 

=  1.7 

1   *7»« 

1.7r 

1.7r 

1.7r 

r 

i  .  /r 

1.7r-l 

log  — 

m 

1.  8 

3.06 

2.  06 

1.  485 

.  17184 

.  22339 

1.672 

2. 

3.4 

2.  4 

1.4166 

.  15124 

.  196612 

1.  572 

2.  2 

3-74 

2.  74 

1.3649 

.  13510 

.17563 

1.  498 

2.  4 

4.O8 

3.08 

1.  3246 

.  122084 

.  I587o 

1.441 

2.  6 

4.  42 

3-42 

1.  2923 

.  11139 

.  144807 

1.395 

2.8 

4.76 

3-76 

1.2659 

.  102O5 

.  13266 

1.357 

3.2 

5.  44 

4.  44 

1.  22522 

.  O8820 

.  11466 

1.  302 

3.4 

5.  7s 

4.  78 

1.  2092 

.  08249 

.  107237 

1.  280 

3.  5 

5.95 

4.95 

1.  2O20 

.07997 

.  103961 

1.  2704 

The  values  in  these  tables  have  been  plotted, 
with  the  chart,  fig. (12). 

The  ordinates  in  this  chart  give  values  of 
n  corresponding  to  values  of  r  in  the  abscissa. 
Each  curve  represents  the  relation  of  m  and  r 
for  a  satisfactory  layout,  with  a  given  air 
column,  the  air  column  length  being  expressed  as 
a  ratio  with  respect  to  the  length  of  recoil. 

The  air  column  lengths  are  taken  at  0.8b, 
l.lb,  1.3b,  1.5b  and  1.7b  where  b  =  length  of  recoil 

Values  of  m  and  r  for  air  column  lengths  inter- 
mediate between  these  values,  may  be  easily  ob- 


613 


-xo'r'*    :  U**v.U  ,?    at 


614 


tained  by  interpolation. 

Solving  for  the  proper  value  of  r,  and  as- 

suming an  air  column  length,  NO  immediately  ob- 
tain ID  and  therefore  paf  since  pai  is  now  known. 

To  prevent  a  rise  of  the  recoil  reaction 
during  the  recoil  at  horizontal  displacement  as 
well  as  to  minimize  the  temperature  rise,  during 
the  recoil,  we  will  limit  m  to  1.8. 

Therefore  r  is  definitely  limited  for  various 
air  column  lengths.   Its  upper  limit  is  more  or 
less  arbitrary,  it  being  desirable  to  prevent  a 
too  bulky  forging  and  obtain  minimum  weight.  The 
upper  limit  of  r  will  be  assumed  at  r  *  3.5. 

Thus,  when  we  have  but  two  cylinders  of  the 
same  length  where  the  air  column  is  somewhat 
shorter  than  the  length  of  recoil  *  0.8b,  we  find  r 
very  definitely  limited  to  a  constant  value  3.5. 

(8)  Strength  of  Cylinders: 

Strength  of  cylinders  should  be  based  on 
maximum  pressures.   As  shown  in  Chapter  IV. 


3 

Dt  =    •    Do  where  pt*  -  x  elastic  limit  of  the 
Pt~P  material  used  (Ibs. 

per  sq.in) 

p-  maximum  pressure  in 
cylinder  (Ibs/sq  in.) 

*  Pmax  usually  4500  Ibs.per 
sq.in.  in  recoil  cylinder 

*  paf  final  recuperator 

pressure  in  recuperator 
cylinder 
D  =outside  minimum  diameter  in 

inches. 

D0-  inside  diameter  which  is  given  in  inches, 
Thickness  between  cylinders  should  be 

Pd+Pafda 
w  *  —  —  —  —   where  w  =  minimum  allowable  tnick- 


615 


ness  between  cylinders  (inches) 

d  =  diara.  of  recoil  cylinder  in  inches 
da»  diam.  of  air  cylinder  in  inches 

(9  )  -  Calculation  of  maximum  and  minimum  throttling 

areas. 

Since  all  port  areas  are  constant  multiples  of 
the  maximum  throttling  area,  the  exact  deviation 
of  this  area  is  of  prime  importance. 

From  (6)  we  find,  for  the  throttling  through 
the  regulator  orifice, 


"h   "  TTT7  -  Z~~\      (max.  throttling   area) 
I/O  vpn~Kv  ) 

V*  A  3TT* 

w8  „   K  A  v      (min.  throttling  area) 
175(pa-Kv) 

where  wh  -  the  max.  throttling  area  usually  at 

horizontal  recoil  (sq.in) 
ws  »  the  rain,  throttling  area  at  max.  elevation 

(sq.in) 
A  =  the  effective  area  of  the  recoil  piston 

(sq.in) 
V  -  the  max.  restrained  velocity  =  0.9  V_ 

approx. 

K  =  -   the   throttling   constant 
0.773 

pj,  »  the  minimum  pull,  usually  at  horizontal 

recoil  (in  Ibs) 
ps  =»  the  maximum  pull,  at  maximum  elevation. 

(10)  Layout  of  Recuperator  Forging: 

In  the  layout  of  a  recuperator  forging,  we 
must  decide  as  to  the  arrangements  of  cylinders. 
Depending  upon  the  value  of  "r"*  _^a_  ,  we  have 
three  possible  arrangements: 

4  o  jr.  pi'SKfT'O  r  '  s<  j 


616 


(1)  Three  cylinders,  the  two  re- 
cuperator cylinders  symmetrical  with 
the  recoil  cylinder. 

(2)  Two  cylinders,  the  recuperator 
cylinder  having  an  overhang  with 
respect  to  the  recoil  cylinder. 

(3)  Two  cylinders,  the  recoil  and  re- 
cuperator cylinder  being  of  the  same 
overall  length. 

Paf 
From  the  chart  giving  values  of  n  =  for 

values  of  r  =  *a  for  different  air       Pai 
colunn 

lengths,  we  see  that  the  values  of  "r"  for  the 
curves  giving  lengths  of  various  air  columns  are 
limited  on  the  one  hand  by  the  maximum  value  of 
m  =  1.8  consistent  with  stability  at  zero  degrees 
elevation  and  normal  use  of  temperature  in  the  re- 
cuperator, thus  giving  the  various  mini  Bum  limit- 
ing values  of  "r"  for  vari  ous  air  columns,  while 
on  the  other  hand  the  maximum  allowable  value  of 
r  =  3.5  depends  upon  proper  counter  recoil  function- 
ing and  layout  considerations.   If  now  r  is  obtained 
by  the  formula, 

r  *  .0309 

The  possible  lengths  of  air  columns  consistent  with 
the  limitations  are  determined. 

If  r  from  the  above  equation  is  low,  then 
we  must  have  longer  air  column  lengths   and  there- 
fore usually  three  cylinders,  whereas  if  r  is  large, 
short  air  column  lengths  are  possible  and  two 
cylinders  may  be  used.   With  arrangement  (3),  r 

becomes  practically  constant  and  unless  r  from  the 
above  equation  falls  in  the  neighborhood  of  3,5, 
it  will  be  necessary  to  increase  the  effective 
area  of  the  recoil  cylinder  with  a  consequent 
larger  recoil  cylinder. 


617 


Having  decided  upon  the  arrangement  and 
number  of  cylinders  from  a  consideration  of  the 

proper  air  column  consistent  with  "r"  we  have  now 
to  obtain  the  exterior  dimensions  of  the  forging. 

Exterior  Dimensions: 

The  primary  exterior  dimensions  of  importance 
are: 

(1)  A  cross  section  of  the  recuperator, 
giving  the  location  of  the  piston  rod 
with  respect  to  the  center  line  of 

the  bore,  the  axis  of  the  several 
cylinders,  and  the  position  of  the 
guides,  thus  determining  the  external 

8-iv   .  }  -  •- if  •  ^*v  ->  -  -    '  T»; 

contour  of  the  cross  section  of  the 
»d  V,*n  lu 

forging. 

(2)  A  longitudinal  section  of  the 
recuperator,  giving  the  overall 
length,  location  of  the  trunnions, 
elevating  arc,  etc. 

In  a  satisfactory  exterior  layout,  the  follow- 
ing points  must  be  observed: 

(a)  The  center  of  gravity  of 
the  recoil  parts  should  be 

made  in  a  vertical  plane  through 
the  axis  of  the  bore  and  at  a 
minimum  perpendicular  distance 

below  the  axis  of  the  bore  con- 
sistent with  a  satisfactory 
layout. 

(b)  The  center  line  of  the  en- 
trance channel  to  the  regulator 
valve, (that  is  for  the  passage- 
way between  the  recoil  and  re- 
cuperator cylinders)  should 
pass  through  the  center  of  the 
recuperator  cylinder.   Preferably 
the  center  line  of  the  entrance 
channel  should  be  in  a  horizontal 


618 


(c) 


plane. 
If  < 

the  connecting  channel  cross 
section,  and  D  the  diameter 
of  the  recoil  cylinder,  the 

distance  between  the  center 
line  of  the  recuperator 
cylinder  and  recoil  cylinder 
must  not  exceed  D 
To  meet  condition 
(a)  the  recoil  axis  is 
usually  nearer  to  the  axis 
of  the  bore. 

To  overall  lengths  of  the 
recuperator  forging  nay  be 
estimated  roughly  from  the 
following  table: 


aa   inches. 
2 


4.  7"-M«  19O6  94"  "TO" 

4.  7  "-«.  1918  68.75"  40 

3.3'-w.l913  86*  60' 

75    »/»-M.19l6  72.83"  46' 


1.52 


Therefore  ordinarily  the  total  length  of  re- 
cuperator forging  trill  be  taken  at  1.5  the  length 
of  max.  recoil.   It  should  be  shorter  if 
practicable. 

(e)     Without  a  balancing  gear, 
for  guns  of  moderate  elevation, 
the  trunnions  should  be  located 
•in  the  horizontal  direction  at 
the  center  of  gravity  of  the 
tipping  parts  plus  one-half 
weight  of  projectile  and  charge 


619 


when  the  gun  is  in  battery. 
More  or  less  error  in  the 
location  of  the  trunnions  as 
respects  the  center  of 
gravity  of  the  tipping  parts 
in  the  vertical  plane  per- 
pendicular to  the  axis  of  the 
bore  will  not  effect  the 
balance,  unless  the  angle  of 
elevation  is  considerable,  and 
the  center  of  gravity  of  the 
tipping  parts  is  considerably 
above  or  below  the  trunnions. 

Therefore,  in  order  to  prevent 

%  :  T-  j  ,  r 

a  reversal  of  the  reaction  on 
the  elevating  arc  and  pinion 
during  recoil  and  counter  re- 
coil, it  is  highly  desirable 
with  guns  of  moderate  elevation 
to  locate  the  trunnions  on  or 
below  the  center  of  gravity 
of  the  recoiling  parts  which 
are  usually  below  the  axis  of 
the  bore. 

When  a  balancing  gear  is  introduced,  as  is 
sometimes  necessary  when  the  gun  fires  at  high 
elevation,  the  trunnions  are  placed  axially  or  in 
a  longitudinal  direction,  farther  to  the  rear  in 
order  to  have  as  long  a  recoil  as  possible  at  max. 
elevation.   Further  with  a  proper  design  of  the 
balancing  gear  location  of  the  trunnions  with 
respect  to  the  center  of  gravity  of  the  tipping 
parts  in  a  direction  perpendicular  to  the  axis 
of  the  bore  is  no  longer  so  restricted  except  that 
in  order  to  avoid  reversal  of  stresses  on  the 
elevating  arc  it  is  desirable  to  locate  the  trunnions 
slightly  below  the  center  of  gravity  of  the  re- 
coiling parts,  but  the  distance  must  be  quite  small 
or  the  arc  reaction  will  become  large. 


620 


In  gone  designs  it  may  be  necessary  to  locate 
the  center  of  gravity  of  the  recoiling  parts  above 
the  bore,  and  the  ponder  pressure  couple  will  then 
be  in  the  opposite  direction. 

If  P^e  is  the  powder  pressure  couple,  and  K 
the  resistance  to  recoil  and  S  the  distance  down 
from  the  axis  of  the  bore  to  the  center  of  the 
trunnions,  in  order  that  there  be  no  reversal  of 
stress  on  the  elevating  arc,  we  oust  have 

(Pb-K)e 

K(S+e)  =  >  Pbe    hence  S  *  >  

In  K 

determining  the  final  values  of  e  and  S,  the  weight 
components,  out  of  battery  and  conditions  existing 
in  counter  recoil  must  be  considered. 

(f)     With  guns  above  155  m/m, 
two  separate  recoil  systems 

*%  L  d  "'• 

symmetrically  placed  above 
and  below  the  gun  should  be 
used.   The  gun  should  recoil 
in  a  sleeve  and  the  trunnions 
should  be  located  slightly 
below  the  axis  of  the  bore. 
Interior  Dimensions; 

The  primary  interior  dimensions  of  importance, 
are: 

(1)  The  port  area  or  channel  leading 
from  the  regulator  towards  the 
floating  piston  in  the  recuperator 
cylinder  should  bear  a  constant 
ratio  to  the  maximum  opening  of  the 
valve  which  occurs  for  minimum  pull, 
usually  at  horizontal  elevation. 

If  «c  »  the  constant  channel  area  from  the 

regulator  valve 
wh  »  the  max.  recoil  orifice. 
Then  wc  =  4.3  wh 

(2)  The  area  of  the  channel  or  port 
connecting  the  recoil  and  recuperator 


621 


cylinder,  wa  should  have  the  fol- 
lowing relation  with  respect  to 
wjj,  that  is,  wa  =*  3.5  to  4.3  wb 

(3)  The  entrance  channel  to  the 
regulator  valve  froa  the  recoil 
cylinder  a,  which  is  also  the  area 
at  the  base  of  the  regulator 
valve,  should  be: 

a  =  k  «h  where  the  limits  of  h  are  2.3  to  3.5 

If  we  pass  a  cross  section  of  the  recuperator 

through  the  center  of  the  regulator  valve,  the 

channel  area  on  either  side  of  the  valve,  that 

is 

wc  -(the  vertical  section  through  the  axis  of  the 

valve  normal  to  the  recuperator  axis  enclosed 

within  the  area  wc )=  w^ 

and   w 
»  ->  c 

If  h  represents  the  depth  of  the  section  wc 
and  da  the  diameter  of  the  regulator  valve  at 
its  base,  we  have  roughly, 
wc  -  dahc  =  0.5  wc  =  0.55  wcapprox . Hence 

"c         wb  "h 

da  '  0.46  =  1.935  —   and  a  =  2.94  

bc         hc  h* 

Mow  in  a  suitable  layout  hc  -  0.2  Da  where  Da  » 
the  diameter  of  the  recuperator  cylinder,  hence 

-& 

a  -  73.5  — 
a 

(4)  The  length  of  the  buffer  chamber 
s  c:  «*;••;••-.  -:i  -    •        '«  tflk 

_n.  ^  ,.    is  based  on  a  consideration  of 

-.•:• 

counter  recoil. 
If  dfa  =  the  recoil  length  during  the  buffer  action 

d£  =  the  length  of  the  buffer 
From  a  consideration  of  counter  recoil, 


622 


7  »rV* 
db(0.15  wr+Rp)» If  V  -  3.5  ft.  per  sec. 

0.8    as  a  maximum  value,  we 
0.238Wr 

have  db*  — — — — - 
0.15Wr+Rp 

*nd  0.238Wr        A 

db'   *    ( ••• )  -T—  *   min*    length   of  buffer 


where  Rp-0.15  K  (d+dr)[  .05*  +  .09(a 

and 

pmax=4500  usually.  The  buffer  chamber  should 
be  made  d£=1.2  to  1.3  d^ 

(11)   Regulator  Dimensions: 

Referring  to  fig.  (11)  let, 

a  =  area  at  base  of  regulator  valve  (sq.in) 

at=  area  of  upper  and  lower  valve  stem  (sq. 

in)  •*«< 

da»  diameter  of  regulator  valve  at  base  (in) 

^•ta  diameter  of  regulator  valve  at  stem  (in) 
c  -  effective  circumference  at  base  of  valve 

(in)                  wfi 
From  (10)  we  find  that,  a  -  73.5  

and  „  D*a 

h 
da  *  9.675  --       where  wh  »  maximum  throttling 

opening  (sq.in) 

Da  »  diameter  of  recuperator  (sq.in) 
Now  da  *0.6da  approx.  and  at»0.7854d|  =0.2825d| 

hence  at»0.36a. 

The  opening  of  the  valve  is  the  effective  lift 

multiplied  by  the  effective  circumference  of  the 
valve  at  the  valve  seat.   Extension  guides  or  "flaps" 
to  ensure  proper  seating  of  these  valves  reduce 
the  effective  circumference  at  the  valve  seat. 
It  is  customary  to  use  three  flaps  of  a  circumferent- 
ial length  each,  equal  to  the  arc  of  60°  angle,  de- 
creased by  two  millimeters  on  either  side,  making 
the  linear  length  of  flap  atthe  circumference  equal 
to  the  arc  of  60°  minus  4  millimeters.   Hence 


623 

nda    12 

c  *  — -  +  " 

2    25.4 

-0.3925da+0.4725 

In  the  throttling  or  lower  valve  and  its  stem 
equalizing  pressure  ports  should  be  bored  within. 
In  the  stem  itself,  the  inside  diameter  or 
diameter  of  the  vertical  port  should  be 

dl  »0.5  to  0.6  d. 
t  *t 

Pour  equalizing  holes  just  above  the  seat  in 
the  regulator  valve,  in  a  horizontal  plane,  meet- 
ing at  a  common  opening  at  the  center  should  be 
inserted.   From  the  center  opening  there  should 
be  a  very  small  vertical  opening  leading  to  the 
recoil  cylinder,  this  acting  as  a  pressure 
equalizer  between  the  recoil  and  recuperator 
cylinders.  The  opening,  however,  should  be  made 
negligible  as  compared  with  the  throttling  open- 
ing and  small  as  compared  with  the  counter  recoil 
constant  orifice. 
(12)  Reactions  on  Regulator  Valve: 

Let  Pb=  reaction  of  Belleville  washer  on 

regulator  valve  (in  Ibs) 
<  reaction  of  spiral  spring  on  regulator 

(in  Ibs) 
p  =  pressure  in  recoil  cylinder  (Ibs/sq. 

in) 
pa=  pressure  in  recuperator  (Ibs/sq.in) 

a  =  area  at  base  of  valve  (sq.in) 

a  -  area  of  valve  stem  (sq.in.) 

h  *  lift  of  valve  from  initial  opening 

(in) 
h0=lift  of  valve  from  seat  of  initial 

opening  (in) 
c  =  effective  circumference  at  base  of 

valve  in  inches. 
Sb=  spring  constant  of  Belleville  washer 

(Ibs/in) 
Sg*  spring  constant  of  spiral  springs  (Ibs/ 

in) 


624 


hjj  =  initial  compression  of  Belleville  washer 

at  initial  opening  (in) 
h  ^initial  compression  of  spiral  spring  at 

initial  opening  (in) 

Then,  at  short  recoil,  or  intermediate  recoil,  we 
have  pa  -p^(a-at  )=Fb+Rs  (approx  )  hence 

(p-pa)a+PaatsRb+Rs  <!> 

and  at  long  recoil,  we  have 

(p-pa)a=Rs(approx)  <2) 

•   :  .•  /• 

Further,  we  have  the  following  lifts  of  the 
, 
valve, 

K  A  V    —          (3)   At  short  recoil 

I  loom 

U)   At  long  recoil 

where  ps  and  ph  are  the  values  of  p  at  short  and 
long  recoil  respectively, 

0.773 

The  spiral  spring  should  be  designed  on  the 
following  basis: 

(1)  The  maximum  compression  should 
be  taken  at  from  2/3  to  3/4  the 
solid  load  of  the  spring. 

(2)  The  initial  compression  should 
be  taken  at  from  1/4  to  1/3  the 
solid  load  on  the  spring.   Hence, 
using  the  maximum  limits,  the 
compression  from  free  to  solid 
height  hfs=2b     (5)  and 
therefore    ,f  n« 

S3 

h  =  -  n  (6) 


where  f83  max.  allowable  torsional  fibre  stress 
(Ibs/sq.in)  (Usually  =  120,000  Ibs.per 
sq.in) 


625 


Ds=  diam.  of  helix  in  inches. 

dg=  diam.  of  wire 

N  =  torsional  modulus  (taken  at  12,000,000 
Ibs/sq.in) 

n  =  number  of  coils  of  the  spring 

Proa  previous  design  layouts,  the  total  height 
of  spring  column,  at  assembled  height  should  not 
exceed  pa   inches.   Hence  the  solid  height  H0 

2°. 

becomes,  —  -  -  h  =  HQ  but  H0=dg(n+l)   hence 

0.5Da-h 

d  =  -  r^—  (7) 

n+1 


Combining  (6)  and  (7)  we  have 
nfD         0.5D-h 


n 


(8) 


2N"  h          n+1 

The  load  at  assembled  height  =  -  Rg 

9 

hence     3nf,d| 

R  =  -  ^—  (9) 

32D 
Combining  with  (7),  we  have 

3nf_  0.5Da-h 


and  with, 

nfa  D«n   0.5D  -h 


(Eq.8)  we  may  determine 


2Nh        n  +  1  ,  _. 

n  and  D.   The 


solution  may  t>e  simplified  by  assuming 

-,  as  a  first  approximation,  we 
u  art*  lo  aoUo» 


0.5Da-h    0.4Da 

^^— — ——  •=  — — — 

0+1         n     then  have 


fsDt 
RS«  .01885 


0.8  Da 


626 


Solving  for  D,  d,and  n,  we  have 


2.62  /  — : inches 


If  we  assume  N=10,500, 000  Ibs/sq.in. 

fs  =  120,000  Ibs/sq.in. 
then 


0.216 


If  D  is  too  great  for  a  satisfactory  layout, 
we  may  increase  the  height  of  the  spring  column 
slightly  or  let  the  maximum  working  load  on  the 
spring  move  closely  approach  the  load  at  maximum 
compression. 

Solving  for  the  diameter  of  wire  "d"  and  the 
number  of  coils  "n",  we  have 
s 


and  if  fs=120,000  Ibs  per  sq. 


in.  then 
d  =  .0305  y  R,,DS 

o   o 

Test  pressures  are  usually  at  double  the 
service  pressure,  hence  the  material  will  be 
strained  up  to  3/4  the  elastic  limit. 

(13)  Design  of  Cam  Mechanism  and  Layout; 

Briefly,  the  action  of  the  cam  is  to  control 
the  motion  of  the  upper  valve  stem  which  reacts 
against  the  Belleville  washers.   At  long  recoil 
the  valve  displacement  (i  .  e.  the  displacement  of 
tbe  unoer  valve  stem)  is  sufficient  so  that  the 
lower  valve  stem,  is  never  brought  into  contact 
with  the  upper  stem,  and  the  lower  stem  is  controlled 
entirely  by  the  spiral  spring.   At  an  intermediate 


627 


recoil,  the  lower  stem  is  brought  ultimately  in 
contact  with  the  upper  stem  and  the  valve  is  con- 
trolled by  the  compound  characteristics  of  the  two 
stems.   As  the  upper  stem  initial  position  is 
brought  closer  and  closer  to  the  lower  valve  stem, 
the  valve  opening  depends  more  on  the  characteristics 
load  deflection  slope  of  the  Belleville  washers. 
Finally  at  short  recoil,  where  the  upper  valve  stem 
is  brought  into  initial  contact  with  the  lower  valve 
stem  and  the  displacement  of  the  cam  is  zero,  the 
valve  opening  depends  practically  on  the  Belleville 
characteristics  alone,  the  effect  of  the  spiral 
spring  being  negligible.   It  is  to  be  noted  that 
the  throttling  at  intermediate  recoils  approximates 
that  if  a  constant  orifice,  with  however  the 
characteristic  peak  effect  in  the  braking  with  a 
constant  orifice  eliminated.   The  throttling,  there- 
fore depends  upon  the  displacement  of  the  valve 
and  the  characteristic  load  deflection  curves  of 
the  Belleville  and  spiral  springs. 
Let  g  =  ratio  of  cam  movement  to  valve  movement 
(usually  taken  at  5) 

X  =  distance  valve  should  lift  to  engage 

Bellevilles  (in) 

hs-  initial  compression  of  spiral  springs  (in) 
bo  =  clearance  of  valve  (in) 
h  =  lift  of  valve  (in) 

hb3  initial  compression  of  the  Bellevilles  (in) 
Sb3  change  in  load  per  unit  deflection  of  the 
Belleville  washers, i.e.  the  Belleville 
spring  characteristic  (Ibs) 

Sg*  change  in  load  per  unit  deflection  of  the 
spiral  spring,  i.e.  the  spiral  spring 
characteristic  (Ibs) 

Then  at  an  intermediate  recoil,  the  reaction  of 
the  spiral  spring,  becomes,  Rg=Sshs+Ss (h+ho)(lbs) 
The  reaction  of  the  Belleville  washers  becomes 
Rb=Sbbb+Sb(h+h0-X)  (Ibs)  while  the  hydraulic  re- 
action becomes,  (p~Pai)a+Pai  at  (Ibs) 


628 


where  p  =  the  intensity  of  pressure  in  the  recoil 
brake  cylinder  (Ibs/sq.in) 

pai  =  the  intensity  of  pressure  in  the  re- 
cuperator (Ibs/sq.in) 

a  =  area  at  base  of  valve  (sq.in) 

at-  area  of  valve  stem  (sq.in) 
Then  for  equilibrium  of  the  valve, 
Ss(hs+h+h0)+Sb(hb+h+h0-X)-[  (p-pai>a+Paiat}*  ° 
Therefore,  for  the  distance  of  valve  lift  to  engage 
Bellevilles,  is 


]> 
J 


X  =  —  JsgOvho+hJ+Sba^+bo+hM  (p-pai)a+paia 

Sb  L 

The  variation  of  the  length  of  recoil  against  ele- 
vation may  be  made  in  any  arbitrary  way,  but,  how- 
ever, the  following  method  is  usually  employed. 

In  general,  assume  the  le  ngth  of  recoil  that 
of  horizontal  recoil  from  0Q  to  #t  degrees,  (usually 
from  0°  to  20°  elevation),  then,  decrease  the  re- 
coil proportionally  with  the  elevation,  (i.e.  from 
20°  to  max.  elevation,  the  recoil  length  decreases 
uniformly  to  short  recoil  at  maximum  elevation). 
Thus  if, 

b  =  length  of  an  intermediate  recoil 
0  *  corresponding  angle  of  elevation 
bh=  length  of  recoil  at  horizontal  elevation 
bs=  length  of  recoil  at  maximum  elevation 
Of  Q-  maximum  elevation 
0^=  initial  elevation  where  the  recoil  is 

shortened 
.  , 
then     b  b 

b  =  -  «?.-0)+bs  (ft) 
0m-<*i 

The  resistance  to  recoil  corresponding  to  the 
length  of  recoil  "b"  is  given  by: 


K  =   1.03[ 


2g                uVf 
b+ (.096+. 0003d) 

a  u 


629 


where  *  -  weight  of  projectile  (Ibs) 

W  »  weight  of  powder  charge  (Ibs) 
*  Wr=  weight  of  recoiling  parts  (Ibs) 
u  =  travel  up  bore  (inches) 
d  =  diam.  of  bore  (in) 
v  *  muzzle  velocity  (ft/sec) 
Vf*  max.  free  velocity  of  recoil  (ft/sec) 

and  *v+W4700 

Vf»  -      (ft/sec) 

*r  0.47WrV| 

For   a   rough    approximation,    K  =   —  —  —  —     (Ibs) 

gb 
The   required   recoil  braking    is  given  by 


(K+Wrsin 

B  =  -  —  Rn  or  B  =  —  -  -  —  --  R_ 
2ueb  l+2ueb 

approximately 
where  1  =  distance  between  guide  clips  (in) 

eb  3  distance  from  center  line  of  bore  to 

center  line  of  brake  cylinder  (in) 
r  =  aean  distance  to  guide  contact  (in) 
R  =  brake  cylinder  packing  friction  (lb») 

For  the  lift  of  the  valve,  we  have 

a 

.098A*V,. 

V.in)  where  A  =  the  ef- 

fective area  of 
recoil  piston  (sq.in) 
Kv  =  recuperator  reaction  (Ibs) 
Vr  =  velocity  of  retarded  recoil,  about  0.9Vf 

(ft/sec) 

c  =  effective  circumference  of  lower  stem  (in) 
we  may  also  express  the  lift  in  terms  of  the 
pressures,  then      >098A  VF 

h  *  -  Un) 


where  p  =  -  a  the  pressure  intensity  in  the  brake 
A   cylinder  (Ibs/sq.in) 


630 


SPRING*  &  CAM   MOTION  (x)  DIAGRAM 
Fig.  13 


631 


Ky 

Pai=  -—  =  the  initial  recuperator  pressure 
intensity  (Ibs/sq.in) 

(14)  Counter  Recoil  Design: 

The  function  of  the  counter  recoil  buffer  is 
to  reduce  the  pressure  in  the  recoil  cylinder  to 
a  very  low  value  practically  zero.  The  recoil- 
ing parts  are  therefore  Drought  to  re^st  by  the 
combined  packing  and  guide  friction  in  a  displace- 
ment corresponding  to  the  buffer  length  in  the 
recuperator  cylinder.   For  a  preliminary  design 
layout,  the  entrance  velocity  into  the  buffer  may 
be  taken  at  a  counter  recoil  velocity  of  1  meter 
or  3.28  ft/sec,  but  preferably  less  than  this.   To 
allow  for  a  margin  in  variation  of  counter  re- 
coil friction  the  buffer  displacement  will  be  re- 
duced in  the  counter  recoil  to  0.7  its  actual 
value.   Then,  we  have 

0.7db(0.l5Wr+Rp-)=  £  ^-^  3728*   hence 
0.238Wr 


The  corresponding  displacement  of  the  buffer  in  the 
recuperator,  is 

A 

—  db   where  A  =  effective  area  of  recoil  piston 

*a  Aft»  cross  section  area  of  recuperator 

cylinder 

The  length  of  the  buffer  rod  will  be  male  about 
20*  greater.   Hence  for  the  length  of  the  buffer 
rod,  we  have 

0.238W-     A 


The  length  of  buffer  chamber  is  usually  constructed 
from  20  to  30*  greater  than  the  buffer  rod,  hence 
d£  -  1.2  to  1.3  db'  (ft)  for  length  of  buffer  chamber 


632 


The  ID  ax  i  BUB  allowable  counter  recoil  velocity  at 
borisontal  elevation  should  not  exceed  3.5  ft/sec. 
The  counter  recoil  velocity  for  a  satisfactory 

design  ranges  from  2.5  to  3.5  ft/sec.  The  velocity 
used  in  counter  recoil  should  be  such  that  with 
the  expression  g 

0.7  ib<0.16Ir.B{>.  ^  55=15.. 

db  ranges  from  1/4  to  1/3  the  short  recoil  bg 

The  packing  friction  for  the  recoil  may  be 
expressed  by  the  relation,  R  -C^+C^p  (Ibs)  where 
p  =  Ibs/sq.in.  in  the  recoil  cylinder.   On  counter 
recoil  during  the  buffer  action  p  =  0  approx.  hence 
R'»Ct  approx.   Now  Ct  is  that  part  of  the  packing 
friction  due  to  the  Belleville  compression  of  the 
packing  and  is  designed  for  the  maximum  recoil 
pressure  paax  (Ibs/sq.in) 

If  Dr=  outside  diameter  of  packing  ring,  (in) 

dr=  diameter  of  rod  (in) 

a  *  depth  of  silver  flange  of  packing  (in) 

a1  «  depth  of  outer  silver  flange  (in) 

b=  packing  contact  (in) 
then  t 

R«Ct*  R(Dr+dr)[.05b  +  .09(a+  —  )]0.15  p>ax  (Ibs) 


The  guide  friction  on  counter  recoil  may  be  taken 
at  RJ-0.15  to  0.2  Wr  (Ibs) 

For  the  total  recoil  friction,  we  have 


Constant  Orifice  Opening:  at  max,  elevation, 

*  A«  V 

(sq.in) 


13.2/paA-Rf-Wrsin<Zf. 

-  at  horizontal   elevation. 


633 


Pai+Paf 
where  p^  =  -  -  -  =  mean  air  pressure  (Ibs/sq.in) 

2 

Pal  =  initial  air  pressure  (Ibs/sq.in) 
pafj  final  air  pressure  (Ibs/sq.in) 
The  orifice  to  be  used  should  be  taken  the  mean 

of  WQ  and  w"   hence     ••+•" 

o  o   (   .  , 
w0=  -£—     (sq.in) 

and( 

2»5  to  3 


KAV 


V0f=2002.5  (ft/sec) 
The  buffer  entrance  area  should  be 

/   '      *£ 

.00894   •  • 
Pi  -  H A  V 


•here  V  =  2.5  to  3.5  ft/sec. 

DESIGN  PROCEDURE  FOB  ST.   CHAMOBD  RKCOIL 

Given: 


Clam. of  bore(incbes)  d  = 
Muzzle  velocity  (ft/sec )  v 
Wt.of  chargeCin  Ibs)  if  = 


Travel  of  shot  up  bore(inches )  u  = 
Max. angle  of  elevation  0m  = 
Min. angle  of  elevation  0  = 


Max. ponder  pressure  on  base 

of  "breech(lbs/sq.in)  Pb  =  24000     Ibs/sq.in, 


Length  of  recoil  at  max. 

elevation  (ft)  be  -  2.5   feet 


634 


Lengtb  of  recoil  at  0° 
elevation  (ft) 


3.75     ft 


w  »  weight  of  charge  (Ibs) 

M  =  weight  of  projectile  (Ibs) 

Wr  *  weight  of  recoiling  parts 


W 


(los) 

(Probable   weight   of   total 
[mount 
[tfeight  of   trail 


3.25   Ibs, 
33.    Ibs, 


1260. 

2700. 
300. 


Ibs 

Ibs 

Ibs 


3000. 


STABILITY  LIMITATIONS. 


Max. free  recoil  velocity 
(ft/sec) 

wv+w  4700 


Max. recoil  reaction 
R(approx)  = 

wr  v« 

0.45  —  —  = 

«  bs 


33.xl500-t-470Qx3.25 
1260 

51.50  ft/sec. 


0.45 


1260  x  51.50 


32.2x2.3 
18,700  Ibs. 


Height  of  axis  of  bores 
above  ground (assumed) 
(ft)  h   = 


3.  ft 


Max . allowable  horizontal 
recoil  (ft) 


bn  max 


Max. velocity  of  con- 
strained recoil (ft/sec) 
Vr*0.9  Vf(approx)  » 


2x32.2 


11.1  ft. 


0.9  x  51.5  =  46.4  ft. 


635 


Recoil  constrained  energy 
(ft/lbs)A  = 


1260  x  46.4 
2  x  32.2 


42,000  ft/lbs 


Recoil  displacement 
during  powder  period 
(ft)  " 

,  ,H+0.5*,  ,,33+0.5x3.25,80 

Er=  2.24(— -)  u  =       2'24<-I^6-  ->Ia  - 

0.41  ft. 

Constant  of  stability 
(assumed) 

Overturning  moment 
Stability  moment 

Horizontal  distance  from 

spade  point  to  line  of     2700x81+300x34 

action  of  W3(ft)  lg  =         30QO      '  =  6.35  ft. 

0S  =  angle  of  stability    20° 

d  =  moment  arm  of  over- 
turning force  = 
ht  cos  ef+ds-1  sin  <t  =      36x0.9397+7.5-81x0.3420  = 

13.60  in. 


Horizontal  recoil  con- 
sistent with  stability 
(ft) 

Wgls+WrE  cos  0  + 

bn  a  - 
2Wp  cos  fl 


/(W8ls+Wr 


E  cos 


dA 
4Wrcos(2f(WslsE ) 


3000x6.35x41- 


42000x1.13 
.96 


2x1260x0.9397 


636 


bb  >ax  3.74  ft  used 
bhsax.3.75  * 


3.74  ft 


APPROXIMATE  DimiBIOi  OF  RICOPIRATOH 
FOROIB08: 


Max.  resistance  to  re- 
coiKat  max.  elevation) 
(Ibs) 


K. 


Min.    resistance   to  re- 
coil (at  horizontal 

elev.Mlbs) 


Max.pull (nax.elev. ) 
Ubs) 
P,-K8+»rsin  0-2R 

2R*Wrsin  0m(approx) 


0.46  1260 
2.6  32.2 

18700  Iba. 


51.5 


0.47  1260 
3.75  32.2 

13100  Ibs. 


18700  Ibs. 


51.5 


Min.pulKO8  elev.) 
(Ibs) 
Pb-Kh-0.3Wr  * 


Initial  recuperator 
reaction  (Ibs) 
Ky  -  1.3Wr(sin0m+0.3 

cos  0B)  » 


1.3"1260(. 9848*. 05"0. 1736)  = 
1700 


637 


Ratio  of  recuperator 
cylinder  area 

Effective  area  of  re- 
coil piston 

Aa 
r  S-A  = 


18700 


=  0.039x46.35 


12730-1700 


2.35 


From  chart  -  assume  air 
column  =  1.36 

r»in  «  2-5 


Total  weight  of  recoil 
piston  and  rods  (Ibs) 


30  Ibs 


Effective  area  of  re 
coil  piston  (sq.in) 

If  r  >  r 


min 


A.  =  0.243 


ff 


Corresponding  max. 
pressure  (Ibs/sq.in) 

PS 
P.ax  -    = 


Approx .max. tens! on 
rods  at  horizontal 
(Ibs) 

¥ 


18700+1260x0.3420+ 
.30 
24000  Xl3'4 


27000  Ibs. 


Assumed  max. fibre 
stress  (Ibs/sq.in) 


max 


»  I  elastic 

limit 


60,000  =  40,000  Ibs/sq.in 


638 


Area  of  recoil  rod  (sq.in) 


max 


Diani.  of  recoil  rod 


dr  »  1.127 


Total  area  of  recoil 
cylinder  (sq.in) 
Ar  «  A+Aa  * 


Inside  diam.  of  recoil 
cylinder  (inches) 
Dr=1.127 


Area  of  recuperator 
cylinder  (sq.in) 
A.  =  rA  * 


27000 
40000 


.676 


1.127  /.676  =    .925   in, 
use   1   inch. 


4.16+0.676  =    4.836  sq 
in. 


1.127  /4.8S6  »   2.48 
inches. 


2.5    x    4.16  *    10.40   sq, 
in. 


Inside  diam.  of  re- 
cuperator cylinder(in) 
D=  1.127 


1.127  /10.40  =3.63 
inches  . 


COMPUTATION  OP  PACKING  FRICTIONS, 


Recoil  friction 


Width  of  leather  contact 
of  packing  (assumed ) (in) 
b  »  0.18  in.  to  0.25  in.»  0.21  inches 


Depth  of  one  silver 
flange  of  packing  cup 
(in)  a  *  0.14  in. to  0.16 
ia.  - 


0.14  inches. 


639 


Depth  of  outer  silver 

flange 

a1  =  0.18  to  0.22  in.= 


0.18  inches 


Constant  spring  com- 
ponent of  total  pack- 
ing friction  (Ibs) 
C  -  w(Dr+dr)t.05b+.09 


(a  + 


J 


n (2. 48+1)0. 05x0. 21+0. 09 
(0.14+0.09)0.15x4500  = 
230  Ibs. 


Pressure  constant  for 
fluid  pressure  com- 
ponent of  total  pack- 
ing friction  (Ibs) 
Cf=  n(Dr+dr)[.05b+.09 

(  a  +  |A)]0.73  = 
fi 


n (2. 48+1) [0.05x0. 21+0. 09 
(0.14+0.09)]0.73  - 
0.250 


Total  recoil  packing 
'friction  (Ibs) 


230+0.250 
Ibs. 


x  4500  =  1350 


Floating  Piston  Friction 


Constant  spring  component 
of  floating  piston 
friction  a 

CJ«1.12nDa[  .05b 
Paf  s  GPaf  G  * 


Pressure  constant  for 
fluid  pressure  com- 
ponent of  total  packing 
friction  (l&s)  C'=1.46nD. 

a  « 
[.05b+.09(a+  -;)]  * 

0 


. 63f0.05xO.  21+ 
0.09(0. 14+0. 09)]paf  - 
0.4Paf 


1.46x«x3. 


0312  »0.52 


640 


CALCULATION  OP  THI  D I H K » 8  I 0 N 8  OF  TUB 


RECUPERATOR  FORCING: 


Max.  resistance  to  recoil 
(•ax  .e  lev  at  ion)  (Ib  s  ) 


uVf 

bg+ (.096+. 0003d  

v 


1.05 


1260*51.50 
64.4 


2. 5+ (.096+0. 0003*4. 134) 

19,700  Ibs, 

80*51.5 


1500 


Win.  resistance  to  recoil 
(horizontal  elevation) 
(Ibs) 

-  *rVf 
h    2g 


uVf 
bb+(.096+. 0003d) 


1250*51.5 


64.4    3.75+(0.096 


Maxinum  recoil  packing 
friction  (Ibs) 


packing  friction) 

Kg 

(Paax»4500  or  — 
approx.  ) 


+0.0003*4.134) 


12900  Ibs. 


80*51.5 
1500 


230+0.250*4500*1360  Ibs. 


641 


Distance  between  clip  re- 
actions (inches )  (assumed) 
1  - 


»-; 


60  inches 


gravity  of  recoiling 
parts  to  mean  friction 
line  (inches)  r1  = 


Distance  from  center  of 
gravity  of  recoiling 
parts  to  axis  of  piston 
rod(inches)  d  = 


6.5  inches 


Coefficient  of  guide 

friction,  n=0.1  toO.2  =     0.15 


7.5 


Pull  at  max.  elevation 
(Ibs) 


""I* 


p  max 


l-2nr 


Pull  at  horizontal 
elevation  (Ibs) 


19700+1260*0.8848 

2x0.15x7.5 
1+  

60-2x0.15x6.5 
-  1360  «•  18800 


2nd; 


l-2nr 


Ct  hence 


w  i" 

12900 
2x0.15x7.5 


60-2x0.15x6.5 
11,550  Ibs. 


-  230 


Excess  recuperator  bat- 
tery reaction  constant 
(recuperator  constant) 
K=l.l  to  1.3  =» 


1.2 


642 


Recuperator  reaction  in 
battery  at  max.  elevation 
k(Wrsin0m+Ct) 


1.2(1260x0.9848+230) 


1.1. 2( 


2x0.15x7.5 
60+2x0.15x6.5 
=  1980 


4.16 


Max.  restrained  recoil 
velocity(f t/sec) 

Vr=  0.92  Vf  = 


Patio  of 

Recuperator  area 

Effective  recoil  piston 
area. 


47.40  ft/sec 


1.625  Vr  /- 


2. 625x47. . 


/  18800 
4500(11550 


2.6 


-  1980 


If  r  <  rm^n(see  chart  and 

assume  air  column)          51.6  inches 

Effective  area  of  recoil 
piston(sq.in) 

Ps  18800 

A  =  *  — —  *  4.18  sq.in. 

4500  4500 

Area  of  recuperator 

cylinder  (sq.in) 

Aa  =  r  A  =  2.6x4.18  =  10.87  sq.in, 

If  r  <  rBin(8ee  chart  and 
assume  air  column) 


643 


Effective  area  of  recoil 
piston 


0.2425 

vr 


If  r  >  3.5(Ti»o  short 
cylinders  -  see  chart) 


Effective  area  of  recoil 
piston(sq.in) 

Ps 

A  =  «oo       ^f 


Area  of  recuperator 
cylinder  (sq.in) 

A,  =  3.5  A 


•a 


Horizontal  recoil  pull 

Ph=Kv+. 000912V'  — 2 —  » 
12.25 


Where  length  of  Air  column  is  assumed 


Length  of  air  column 

in  terns  of  length  of  max. 

recoil  (assumed) 


1.445 


Ratio  of 

final  air  pressure 
initial  air  pressure 


M  »(  .  ,)*  '  (see  chart  of 

rj        tables)  =      1.5 


644 


Initial   recuperator 
pressure    (Ibs/sq.in) 

v  1980 

—  »  *   473   Ibs/sq.in. 

A  4.18 


Final  recuperator 
p re ssure(  Ibs/sq.in) 

Paf  *  Pai  (approx)  =    1.5  *  473  »  710  Ibs/sq.in, 


Initial  air  volume  (cu.in) 

*~Al   =  10.87x51.6  =  580  cu.in. 


When  ratio  of  final  to  initial  air  pressure  is 
assumed. 


Assume 

Paf  Paf 

m  =  =  — r-  =  1.5  =   1.5 

Pai  Pai 


Initial  air  volune(cu. 
in) 

0.77  —0.77 

Vo  -  (m".77_  1  >A.  bh»   (4=^^)10.87x45 


560  cu.in. 


Length  of  air  column 
(inches) 


560 


a 


51.8  in. 


A,  10.87 


Initial  recuperator 
pressure  (Ibs/sq.in) 

'   .  ~  »  i222  ,  473  Ibs/sq.in, 

4.18 


645 


Final  recuperator  pres- 
sure 


Paf 


4.73  »  710  Ibs/ 


sq.in 


INITIAL  AND  PIMAL  AIR  PRESSURE  AHD 
AIR  VOLUME. 


Initial   recuperator 
pressure    (Ibs/sq.in) 


473   Ibs/sq.in 


Floating    piston 
friction(initial)(lbs) 

Ct+C»  Pai   = 


.4x710   +0.52  x    473 
530   Ibs. 


Drop   of   pressure   across 
floating   pistondbs/ 
sq.in) 


[*cs   Pai 


ai 


Final  air  pressure 
(Ibs/sq.in) 

Paf  =  rap     z 


Final  air  volume  (cu 
in) 


z  V  - 


o  -  A  bh  = 


473 


530 

10.87 

3-,<m  a  I 


52°  lbs/ 


sq  .in, 


1.5x520  =  780  Ibs/sq 


in. 


560  -  4.18  45  »  370 
cu.in 


646 


Average  drop  of  pressure 
across  floating  piston 
Qbs-sq.in) 


Strength  of  cylinders 


ianiS 

.4x780+0.52(780+520)0.5 
10.87 

»  620  Ibs. 


Test  Pressure  2  X 
Service  pressure. 


Area  of  recoil  cylinder 
Ar=Ar+ar  = 


Diameter  of  recoil 
cylinder(inches)Dr 

1.127  /  Af  = 


Diameter  of  recuperator 
cylinder  (inches) 

Da  -  1.127  /  Aa  = 


4.18  +  0.79  =  4.97 
sq.in. 


1.127  /4.97  =  2.51 
i  nches . 


1.127  /  10.87  =  3.71 
inches . 


Max.  allowable  fibre  stress 
for  cylinders  (Ibs/sq.in) 

Pt  -  -  elastic  limit  = 


Dro  •  Dr 


-  60000  =   22500 


Ibs/sq.in. 


Min.   outside  diam.  of 
recoil  cylinder(incbes ) 


2.51 


22500  -t-4500 
22500-4500 


3.07  inches. 


647 


Min.  outside  diam.  of 
recuperator  or  air  cylinder 


3.71 
Pt~Paf  22500    810 

3.84  inches 

use  3.96  inches. 


Min.    width   between   ad- 
jacent cylinders 
(inches) 


w 


PmaxDr*PafDa  4500x2.3.05+810x3.71 


1.5  pt  1.5  x  22500 

=  0.393 


Calculation  of  max.  and  min.  throttling  areas 

Max.  throttling  area 
(at  horizontal  recoil) 

(sq.in)        I  1 

.098A*Vr  0.098  *   4.18  x47.4 

*h     S  y 1 

(11550-1980)* 
=   0.374   sq.in. 

Min.  throttling  area  at 
elevati 

.098A*Vr 


max.  elevation(sq.in) 


(18800-1980)' 


=  0.282  sq.in. 

„*  «aia  Isc 


LAYOUT  OF  RECUPBBATOR  FORGIMG,  PORT 
AND  CHAKKEL  AR3AS. 


.  of  cylinders  = 


648 


Overall   length  of   forging 

(inches) 

lf-1.5  bh  »  1.5x45  »   67.5   in. 


Diameter  of    recoil 
cylinder(inches)   Dp  »  2.51   in. 

Diameter  of  recuperator 

air  cylinder(incbes) 

Da  -  3.71  in. 


Length  of  air  column 

(inches)  la  »  51.6  in. 


Area  of  connecting 
channel  between  recoil 
and  recuperator 
cylinder  (sq.in) 

w.  »  3.5  to  4.3  wh  =       4.02x0.374  =  1.5  sq.in. 
" 


Diarn.  of  connecting  channel 
(inches) 

dl   »   1.127  /7T  »  1.127     ifi  =    1.38   in. 


Max.  depth  of  recuperator 
cylinder  below  recoil 
cylinder  (inches) 

da  1.38 

0'  »D. *  2.51 »  1.82  in. 

2  2 


Constant  channel  area 
from  regulator  valve 
(sq.in) 

•c  »  4.3  wh  =  4.3x0.374  =  1.606  in. 


Depth  of  constant  chan- 
nel area  Nc(inches) 

hc  »  0.2  Da  =  0.2x3.71  =  0.742 

Extreme  area  to  regulator 
valve  (sq.in) 

a  -  73.5    - 


649 


Diameter  of  entrance  channel 
(inches) 


da  »   9.675  —  = 

Uo 


9.675 


in 


PK8I8K  OF  B10ULATOB. 


Area  at  base  of  regulator 
valve  (sq.in) 

"n 
a  *  73.5 


D: 


Diam.  of  regulator  valve 
at  base  (inches) 

*h 
d.  =  9.675  ~  = 

8        Da 


Diam.  of  upper  and  lower 
valve  stem  (inches) 
da  =  0.6  d0  = 


Cross  section  area  of 
upper  and  lower  valve 
stem(sq.in) 
at  =  0.36  - 


Diam.  of  inside  port  in 
valve  stem 

dl=0.5  to  0.6  d-  - 
a 


0.747  in. 


0.975  in. 


0.6  x  0.975  =  0.585 
in. 


0.36x0.747  =  0.269 
(sq.in) 


0.66x0.585  »  0.322 
(sq.in) 


Guides  or  flaps  at  base  of 
valve  use  =3  subtends  are 
60°-  2mm  on  either  end. 
Length  of  one  flap  at  base 
of  valve  c1  =  0.524  da  -0. 
1573  » 


0.524x0.975  -0.1573= 
0.354  in. 


Effective  circumference  at 
base  of  valve  (throttling 
area)  (inches) 
c  *  1.571  da+0.4725  » 


1.571x0.975+0.4725= 
E.004  in. 


660 


Load  on  spring  and 
Bellevilles  at  short 
recoil  (max.  elevation) 
(Ibs) 


Load  on  spiral  spring 
at  long  recoil  -(0° 
elevation)  (Ibs) 


Lift  of   valve    (inches) 
short   recoil(max. 
elevation). 


.098A2    V, 


(4500-473)0.747*473x0.269 
=*   3140    Ibs. 


4.18 
1710 


-  473)    0.747  = 


h1 


Lift   of   valve(inches ) 
long   recoil(0°  elevation) 

3 

.098   A*    Vr       Vh 

h"   *       y  -  —  * 

°/Ph-Kv 

Load  at  solid  height 
on  spiral  spring  libs) 

R»«  *  r  R.  s 


0.282 
2.004 


0.1405   in 


0.374 
2.004 


0.1862  in. 


1710  -  2280 


651 


Spiral   regulator   spring 

iil?J!^3 

vfax  .  tor  sional    fibre 
stress  (Ibs/sq.  in) 

100,000] 
fs    =   120,000  >   = 

140,000  J 

Torsional   modulus 
(Ibs/sq.in) 

:i?.f*  y 

11,500,000  1 
N  =      10,500,000  f  = 
10,000,000  J 

SeS    ^fii 

-vyfc  ri^i 

H5i     ilOO* 

Diam.of   helix 
spiral    spring 
regulator   valve 

1  *           C  * 

pinches;         »  

f&          3 

/N*h"^R 

;  i  't)  bo' 

io   ci-tgn; 

Ds-   4,.G^/    •;      8 

£< 

If    N   =    10,500,000 
fs     =          120,000 
(Ibs/sq  .  in  ) 

A"*SS 
D  =0.216  /  = 

a                                         3 

D! 

Diam.    of   wire   of   spiral 
spring    (inches  ) 
3/CD7 
i   =1.503   7     ?    ?   = 

fs 

If   fs   =120,000   Ibs/sq.in. 
i   =.0305   •/R3DS 

-  •  5  c  c?  . 

'      -%         A    J     "fcT 
"-*•*/       —    1  9w 

.  « 

652 


COUETiK  BICOIL  DgSIGH  -  BUFFER  DBSIGH 
COHSTAMT  ORIFICg  AMD  PORT 
AREAS. 


Packing  friction  at  end 
of  counter  recoil  (Ibs) 


23°  lbs 


Recoil    length  during   buffer 
action    (ftl 

0.238Hr  0.238x1260 

db   *   r>   isw   +R'°  -     =   °-715ft 

0.15Wr+Rp  .015^1260+230 

l!L"'to'bs)   _8-61n- 

Length  of  buffer  rodlft) 
0.238W 


.        A 

=  1.2  -  -     f     = 
0.15Wr+R^    Aa 

2  db  1.2x.715 

=  -  =    .33   ft.= 

r  2.6 

3.96   in. 


Length  of  buffer  chamber 
(ft) 

d£»1.2   to   1.3   db'   =  1.25   x   3.96  =   4,95   in. 


Win.  allowable  counter 
recoil  velocity  (ft/ 
sec)  at  max.  elev. 


Max.  allowable  counter 
recoil  velocity  (ft/ 
sec)  at  horizontal 
elev  . 

v0.  *  2.5  -  3.5  ft/sec?   2.5  ft.  sec 


653 


Total  counter  recoil  friction 
(nax.  elevation)  (Ibs) 

W_(sin0_+0.3cos0m) 


+0.3Wrcos0m 


230+.25[ 


1260('9848* 


4.18 

.3X.1736, 

]  +.3xl260x 

.1736  =  290  Ibs. 


Total  counter  recoil 

f riction(min.  elevation) 

(Ibs) 


230+.25(^1H^) 
4.18 

+.3x1260=700  Ibs. 


Recuperator  mean  pressure  (Ibs/ 
sq.io) 


473+710 


=590 


Required  constant  counter  re- 
coil orifice  at  max .elevation 

8 


1.25x4.18 


(K=1.25) 


Required  constant  counter 
recoil  orifice  at  0°  elevation 
(sq.in) 


Vol 


13.2/p'A-Wt 


13.2/590x418-290 

-1260X.9848 
.0645  sq.in. 


1.25x4.18   x3.5 
12.2/4.73x4+18-700 
=.0765   sq.in. 


654 


0.0645.0.0765 


Entrance  buffer  area 
(sq.in) 


KAV   /   1 
"b  '  TTT  / 


.00894A2v*   1.33x4.18x3.5 
Pa «  


13.2 


.00894x4.18  *3.5 
590 

(.0705)a 
v  =  3.5  f tXsec(approx) 

Layout  entrance  area  of  buffer,  with  re- 
quired depth  in  groove.  Decrease  depth  of  groove 
to  zero  at  end  of  buffer  du . 


Deflection  from  free  to 
solid  height  of  spiral 
spring  (in) 

hso  s  2h"  *  2x0.1862  *  0.373  in. 

_      _  ____  ___  ___         _      

Spiral  spring  constant 
(Ibs.per  in.  ) 

Rsc  a  2280 

83  "  hs0  3  .373 

This  spiral  spring  will  be  too  bulky  for  practical 
purposes.   Therefore  we  will  let  the  Belleville 
spring  washers  take  care  of  all  conditions  at 
different  elevations  and  design  the  spiral  spring 
strong  enough  to  keep  the  valve  closed  when  gun 
is  in  battery. 

Spiral  spring  reaction 
at  short  recoil 

R;  -s,<k'*h0).       o 

h0«.0197  (initial  lift)(in.) 


655 


Load  on  Belleville  at 
short  recoil  (max.  elev  ) 
(Ibs) 

Rb  3KP~Pai)a+Paia'}~Rs=  (4500-473)0.747-0=  3010 
!  _  lbs' 

Load  at  solid  height  on 
Belleville  washers(lbs) 

Rbo=  IRb  =  f  x  301°  -  4520  Ibs. 

Deflection  from  free  to 

solid  height  of  Belle- 

v  i  lies  (in) 

hbo=3h«  =  3x0.1405  =  0.422 

Belleville  spring  con- 
stant (Ibs.  per  in) 

Rbo  4520 

"  1070° 


compression  of  Belleville  washers. 

n  =  no.  of  Belleville  spring  washers  = 

0.422  _     6 

.071 

hfc=initial  compression      =   115 
h0=valve  clearance  .0197 


Ratio  of 


DB8IGH     OF     CAM     MECHAHISM     AtiD     LAYOUT. 

Cam  Movement 


Valve  movement 

g   =  5 

(taken  usually  at  5) 


In  general,  assume  the  length  of  recoil 
at  horizontal  recoil  constant  from  0j  to  #t  de- 
grees, (usually  from  0°  to  20°  elevation);  then, 
decrease  the  recoil  proportionally  with  the 
elevation,  that  is:-  20° 


656 


Length  of  intermediate  recoil 
(ft) 


45-30 


80-0) +30 


Kg  Total  resistance  to  recoil: 

0.47WrVf  0.47x1260x51.5 


32. 2b 
48800    58700 


Pg2»Total  recoil  pres- 
sure » 


k,+1260sing 
1.1 


-  210 


b  *  lift  corresponding  to  required  throttling 
opening  (inches) 


.098A«V, 


0.178V, 


0.178x47.4 


(p-Pai>* 


8.45 


t 

fc" 

k, 

.in* 

K     +• 

p 

-k 

(p- 

(p- 

h 

sin0 

p*i> 

».!>• 

20 

45-.OO 

13O5O 

.  342O 

13430 

120OO 

10OOO 

2390 

48.  89 

.  1725 

25 

43.75 

134OO 

.  4226 

13930 

12470 

10470 

2500 

50.OO 

.  1680 

25 

41.  25 

142OO 

.5736 

14920 

13370 

11370 

2710 

52.O6 

.  1624 

50 

37.50 

15650 

.  766o 

1662O 

14900 

12900 

3030 

55.  50 

.  1520 

65 

33-75 

17400 

.9063 

18540 

16640 

14640 

3500 

59.16 

.  1425 

75 

31.25 

18750 

.9659 

19970 

17930 

15930 

3800 

61.64 

.  1370 

BO 

3O.OO 

19550 

.  9848 

20790 

18690 

16690 

3980 

63.09 

.133fi 

Linear  notion  of  cam  rod  against  elevation. 
sidering  spiral  spring  reaction  negligible. 


Con- 


h +  h8  - 


(P-pai)a+paiat 


0.175+h- 


(p-pai)0. 747+127 


10700 


657 


a 

b 

o 

d 

e 

f 

X 

Vfax.of 

cam   rod 

inches 

3 

b" 

h  cor- 

h + 

(P- 

3  +  127 

f 

d-g 

rected 

0.175 

0^747 

10700 

20 

45^00 

.  1867 

.  3612 

1785 

1910 

.  1784 

.  1828 

.911 

25 

43.75 

.  1800 

.3550 

1965 

1990 

.  I860 

.  1690 

.  845 

35 

41.  25 

.1110 

.  3460 

2030 

2160 

.  20  20 

.  1440 

.  720 

50 

37.50 

.  1600 

.  3350 

2300 

2430 

.  2270 

.  1080 

.  540 

65 

33.75 

.  1500 

.  3250 

2610 

274O 

.  2560 

.  0690 

.345 

75 

31.25 

.  1450 

.  3200 

2840 

2970 

.  2780 

.0420 

.  210 

80 

30.00 

.  1405 

.3155 

2970 

3100 

.  29OO 

.0255 

.  127 

105   M/M   HOWITZER 


75  M/M  GUN  (Double  Charge) 

MOUNTED  ON  SAME  CARRIAGE. 

Given: 

d  -  diameter  of  the  bore  (in) 

75  m/m  Gun 
Normal 
Super 

105  m/m 
How. 

2.953in. 

4.134in 

v  =  nuzzle  velocity  (ft/sec) 

1500 
2175 

1500 

w  =  weight  of  charge  (Ibs) 

1.401b3. 
S.OOlbs. 

3.  25  Ibs 

u  =  travel  of  shot  up  bore  (in) 

109.50in. 

BO.OOin 

0m=  max.  angle  of  elevation 

80° 

0^=  min.  angle  of  elevation 

0° 

w  -  weight  of  projectile  (Ibs) 

151bs. 

331bs. 

Pjj=max  .powder  pressure  on 
base  of  breach  (Ibs/sq.  in.  ) 

34,000 

24,000 

bg  =  length  of  recoil  at  max. 
elevation 

1.3ft. 

2.5ft. 

t>h=length  of  recoil  at  0° 
elevation 

2.4ft. 
3,75ft. 

3.75ft. 

658 


WBIQRT  Of  GUN  AND  CARRIAGE. 


Similar  Guns 

W 

V 

E=Muzzle 
Energy 

*g 

E/wg 

*t 

w= 
X*  wt. 

75  BB.  French 

16 

1700 

716000 

1050 

705 

2657 

39 

75  mm.  U.S. 

16 

1700 

716000 

750 

956 

3045 

25 

75  mm.  British 

16 

1700 

716000 

995 

720 

2945 

29 

3.8  How.M. 

1906 

50 

900 

378000 

432 

876 

2040 

22.6 

4.7  Gun  M. 

1906 

60 

1700 

2690000 

2688 

1000 

8068 

33.6 

4.7  How.  M. 

1908 

60 

900 

755000 

1056 

716 

3988 

27. 

6"  How.M. 

1908 

120 

900 

1510000 

1925 

785 

7582 

25.7 

155  m/m  How. 

(Sch) 

95 

1420 

2970000 

2745 

1080 

7600 

36.5 

155  m/m  Gun 
(Fil.) 

95 

2300 

8400000 

8795 

960 

25600 

34.5 

155  m/mHow. 
(St.  Cham) 

95 

1520 

3400000 

3040 

1120 

7700 

25.3 

8"  How.  VI 

200 

1300 

5250000 

6652 

790 

19100 

35.0 

8"  How.  VII 

200 

1525 

7200000 

7730 

933 

20050 

38.7 

Average  E/wg  of 


888 

E/wg=1000 
E/wg=888 


v  7 


normal  gun 


super  gun 


howitzer 


15x1500 
64.4x1050 
15x2175 
64.4x1000 

33*1500 

64.4x1000 


1100*    1240 
*1150<(   1290 


1100+30  »  1130 


Wr  How.   1180+30  =  1180 


Average   1155# 


1260# 


659 


Using  highest  %  of  *»r  to  Wt  (39*)Wt  -     -  =  2970* 

397 


2970  -  1155 


1815f 


Wr     Weight  of  recoiling  parts  1230*  and  1260* 
for  gun  and  howitzer  respectively  are  the 
minimum  weight  that  could  be  used  on  account 
of  stresses. 

The  condition  being  to  get  the  minimum 
weight.   These  values  are  used: 

W.  »  1230*  gun 


»r  *  1260*  how. 

Gun 
super 

38.00 

105  m/m 

How. 

51.50 

7 

'.,,.  i.  r,.tL    -in-'  -;    "I--5"  _r  ^       i-ll.inji 

5m/m 

ormal 

23.60 
1 

Vf  max.  free  velocity  = 
wv+4700  w 

15x1500+4700x1.4 

1230 
15x2175+4700x3 

1230 
33x1500+4700x3.25 

1260            " 
sec 

Ks  Resistance  to  recoil 
at  80°  elevation 

wr 

=  1.05[  —  V2* 
2g   r 
1 

uV,1 
b_  +  (.096+.  0003d)  —  *• 
2  v 

1230x23.60 

1500 


660 


•1.05U0640*   1 

1.5(. 096*. 0009)1. 724 


1 .      1.05x10640        fl__- 

'1.05(10640*    )=   *    o720 

1.5+.167  1.662 


_f  1230x38.  00  __  1 
64.4  "X1.5 

1.05x27600 


17600 


1.645 

1260x51.50 


64.4  V 

2 .5+ ( . 096+ . 0003x4 . 134 ) 


1500 

1.05*51900  .  19700 

2.761 

h  =  height  of  axis  of  bore  above  ground. Assumed  36" 

bh  =  max.  allowable  horizontal  recoil  »  /  

2g 

,  /23'60   x3 51",  82  ",1004" 

64.4 

Vr  *  max.  velocity  of  constrained  recoil  ,9Vj(app) 

21.20,34.20,46.35 

A  «  recoil  constrained  energy  =  -£ — 
1230x21.20 


8580 

i 

1230x34.20 


64.4 

22400 


64.4 

1260x46.35 


42000 
64.4 


661 


E  *  recoil  displacement  during  powder  period. 

,  1B  n."^~ 


1230 


12 


1230x12 


.15+.5X3.   109.5   2.24x16.5x109,5 
2'24(        }  -  *  '275  ft 


1230 


1230x12 


33+.5x3.25x  80   2.24x34.63x80 
2.24(  -  )  —  =  -  =  .41 


1260 


12    1260x12 


STABILITY 


ls  - 


3000 


..  6.35.n. 


c  =  constant  of  stability  =  '  «  .96 

stabilizing  moment 

efg  *  angle  of  stability  20° 

d  -  moment  arm  of  overturning  force 

htcos0+d8=l,,sin0=36xO. 9397+7. 5-81x0. 3420  » 

V  O 

13.60   in. =1.13   ft. 
bn   =    length   of    recoil   at   the    angle   of   stability 

¥sls+WrE   cosg±/|*sIs+WrEco8g)*-4Wrcos0(WslsE -) 


662 


3000x6. 35+1260* 


0.4lx.09397+y^Hsl8+WrEcos0)11- 


2x1260x0.9397 


4xI260x  .9397(3000x6.  35x  .41) 


.96 


19, 540*/119, 540) '-4740 (7800-49500) 
2370 


19. 540+/110,000, 000   19.540+10490 


2370 


2370 


3.75  ft, 


RBCUPgRATOR  FOHGIBG8. 


Approximately 


=  maximu!n  resistance  to  recoil 


.45   r 


.45  1230 


.       .     .        nn 
--  VI  *  -  -  23. 
b   g       1.5  32.2 

.45  1230 


„ 
60 


1.5  32.2 


*  38. 


2.5  32.2 


*  n  in.  resistance  to  recoil 
0.47  ^r  y,  m  0.47  J[r 


3.75  32.2 


75  »»  Gun    105  *• 

Mora-  Super  How. 


6380 

16500 

18700 

2660 

663 


• 


0.47   1230 

3.75   32.20 

0.47   1260 
3.75  *  32.2 


-^ 
&1.5 


Pg  =  max.  pull  =  KS  approximately 
Pfi  =  nain.  pull  =  Kh~0.3  Wr 

Kv  =  initial  recuperator  reaction* 
1.3Wr(sin0_+.3cos0_; 

X  ill  til 

=  1.3xl260(. 9848+. 3x. 1836) 
»  1.3x1260x1.037 


r  » 


recuperator  cylinder  area 
eff.area  of  recoil  piston 


A. 

~  »  .039V, 


.039x23 


.60  /— 


2290-1700 


10.1 


Ph-1700 


75  — 
•  or* 

al 

Gun 

-Super 

lOg      KB 

HOB. 

6740 

13100 

•380 

16500 

18700 

2290 

6370 

12730 

1700 

5.35 

664 


_  19000+17150 

Ph  =  3560 

paf    final  air  pressure 

n  '  - —  «  .  .  .  . : (generally)  1.5 

Paji    initial  air  pressure 

e  =  length  of  air  column  from  chart       1.25b 
A.  =  effective  area  of  recoil  piston  = 

ps   19000 


4500   4750 


4.00  sq.in. 


(Usually  packing  is  designed  to  stand  a  pressure 
of  4500  to  5000  Ibs) 

PS  *  max.  pressure  corresponding  to  r  =  3 

P^  =  I222?=  4750  Ib3< 

4 

WQ  =  total  weight  of  recoil  piston  and  rods, 30  Ibs. 
T.  *  max.  tension  on  the  rods  at  0°  elev.»K<-  +  ffr 


30 
19300+13.45x24000*  =27000 

1260   t 
=  assumed  fibre  stress  =  -  elastic  limit 


max 

70000 


«  35000  Ibs/sq.in. 


27000 
Ar  -  area  of  the  recoil  rod  =       =  .772  sq.in. 


35000 
dr  =  diameter  of  recoil  rod  =  1.127 


/ar  =  1.127/772 


+  .99in.(nake  1  in. 

Ar»  total  area  of  recoil  cylinder  =  4. +.781  -  4.781 

sq.in 

Dr  3   inside   dianeter   of   recoil   cylinder  - 
1.127/~Tr   «    1.127/4.781  =    2.46   in. 

Ar   =   rA   *  3x4   =    12   sq.in. 

Da   *    1.127  /TI  =    1.127  /T2  =  3.9  in.    diaicster 

a  a     « 

of  float- 
ing piston. 


665 

CALCULATION  OF  PACKING  FRICTIOH. 

b  =  O.lSin.to  0.25in.use  0.21in. 

a  =  depth  of  outer  silver  flange  of  packing  cup  0.14  in, 

to  0.16  in.,  use  0.14  in. 

a1  =  depth  of  outer  silver  flange  0.18"  to  22", '0.18" 
c t  =  constant  spring  comp.of  total  packing  friction 

a1 

ct  =  n(Dr+dr)[.05b+.09(a+  —-)]0.15  Pmax 

45 

=  n(2.46+.?81;r.05x.2H-.09C.14+.09)).15  Pmax 
»  10.2(.0105+.0207).15Pmax 

=10. 2x.0312x. 15x4750 
c1  =  226 

a1 

C_  =  n(Dn+d_) [ .05b+.09(a+  — ) ] . 73 

JT  I     I  Q 

=  n(2.46+.781)[.05x.21+.09(.14+.09)].73 
=  10. 2x. 0312". 73 
C  =  .232 

2 

Rp  =  total  recoil  packing  friction  -  ct+c  p(p= 

Ibs/sq.in) 

226*. 232  x   4750  1326  Ibs. 

FLOATING  PISTON  FRICTIOH 

*.i>3 
cls  cast(spring  constant)  of  floating  piston  = 

at 

1«  12  ^Do  I  •05b^*»09(.£L +~"~*)  j  p o  f  =  G  P a ^ 

3  '  d  I          o  1 


=  1.12  x  n  x  3.9[.05x.21+.09(.14+.09)]paf=  GPaf 
=  1.12  x  n  x  3.8  x  .Q312Paf 

=  ,428Pa£=GPaf(in  Ibs)  Paf  =  final  air  pressure 

^  =  pressure  constant  for  fluid  pressure  corap. 
of  total  packing  friction. 

- 


=  1.46nDa[  .05b  +  J 

=  1.46nxD  x. 0312=1. 46  x  n  x  3.9  x  .0312 

• 

=  .558 


666 


D2SI6N  OF  RBCUPSRATOR 


75m/m  Gun   105m/m 
Normal   Super  How. 
6720    17600  19700 


b_+(.096+.  0003d) 
Kh  =  min.  resistance  to  recoil 

B  V!       i 


uV 


b  +  (.096+.  0003d)  —  - 
v 


1230x23.60 
64.4 


1500 


10640 


bh+.167 

-  « 
1230x38 


(bh=2.4  ft.  =29  in.) 
1 


27600 
-  - 
3.925 

1260x51.5 


4140 


64.4     3.75  +.267 

^  —  t 

1260x51.5 

- 

64.4x4.017 


max 


226+  .232 


6720 


7030   Ibs. 


12,900  Ibs, 


616   1236   1366 


1  *  distance  between  clip  reactions 


60in. 


-  616 


-  616 


7044 


1.0388 
17600*1210 


-  1236 


16900 


1.0388 

19700-»-1240 
1.0388 


-  1366. 


Pb=pull  at  horizontal  elevation  in  Ibs. 

3560 


667 


r1  =  distance  from  center  of  gravity  of 

recoiling  parts  to  mean  friction  line.   6. Sin. 

n  »  coefficient  of  guide  friction  (.1  to  .2)   .15 

d^  *  distance  from  center  of  gravity  of  recoil- 
ing parts  to  axis  of  piston  rod        7. Sin. 

PULL  OH  THB  ROD 

75  m/m  Gun 
Normal    Super 

Pg=pull  at  max.  elevation 
K8+Wrsin0 

2nd"L 
1-2  nr 

6720+1230X.9848 

—  ___^_^_— _ 

2x.l5x7.5 
60. 2x. 15x6. 5 

6720+123QX.9848 
1.+.0388 

7930 


18800 


2ndK 
l-2nr 


-  C 


668 


• 

p  ,(  -  D  --  226).  945 
1.0388 

(3560x1.  0385+226)  1.0388=Kh=4140 
7030 


1.0388 


-  226)x.945  6180 


12900 

Ph=(  --  226)*.  94.5  11550 

1.0388 

R  *  excess  recuperator  battery  reaction 

constant  1.2 

(1.1  to  1.3) 

Kv*  recuperator  reaction  in  battery  at 
max.  elevation 


R(WriinBm*c  )      1.2(1260«. 9848*226) 
KT.  ,  


2w    "1  C  „  *7   C  OOO 

1-R( +_^j      l  1>s^  *.1&*7.5     ^  .233 

1+2nr   A  60+2x.l5x6.5     4 

1.2x  1466  1760     1760 

___^______^__^___   —  _______  —  ____^^_ 

1-1. 2(. 0363+058      1-.1128   .8872 

Ky  a  1980 

Vr  -  .92Vf=. 92x23. 6=21. 7=Vr;  35.00  =  47.40 


_  _  o  enc  tr    / S _   o  coc^oi   "  /    188QQ 


AC 

r  -  —  -   2.625  Vr  / '-* »    2.625x21, 

A  P    (Ph-Kv)  4750(3560-1980) 


57 


4750x1580 
r   «   2.85 


669 


r.la  .  2.625  Vr      168°°  .  81.6 


4750(6180-1980)  4750*4200 

=  2.79 


af 

-  1.5  =  B 


1  -  l.Sb 

ps  18800 

A  -  »  -  -   3.96 

4750  4750 


Aa  =  rA  =  2.85x3.96=11.30  sq.in.»Aa 

Pjj=nin.  pull  on  the  rod 

Vr  =  velocity  of  recoil  corresponding 

R  »    1.295 


w?  =U0373  rA>2 


h  "v 

r  .2435 

175x. 00139 (Ph-Kv) 

Pp  =  pressure  the  packing  should  withstand 


P*V« 
6.9  — S-1- 


Pp(Pn-Ky) 


r  »  2.625  V. 


j    =    length  of   air  coluran   in   terns   of   max.    recoil 
1.        1.3b 


670 


j  -  1.3 

•  *(  r'1   )  '    from  the  chart  1.5 
r.J-1 

Ky    1980 
paj  =  initial  air  pressure  (Ibs/sq.in)  =  -—  *  — — 

A      o  »  <7w 

Pai  *  500  Ibs/sq.in.  (approx.) 

Pif  *  *  Pai  =  !-5  x  500  s  75°  x  paf  approx. 

VQ  *  initial  air  volume  =  Aa  x  ia  =  11.30  x  45  » 

662  cu.in.  =  V& 


-  1 


INITIAL  AMD  FINAL  AIR  PRBSSURB  AMD 


AIR  VOLUME. 


50° 


Rt  »  floating  piston  friction  initial 


3  Ci+C«  Pai  =  -428xPaf  +.558^500 

=  .428x700+.  558x500 

=  321+279 

Bt  «  500 

P   »  Pli  +  -1  -  2  ai   =  500  =  -  »  500+44.2 
ai          Aa  11.30 

Pai  -  544  Ibs/sq.in.    Pai  =  540  Ibs/sq.in. 

paf  =  *  Pai«l.  5x544=816  Paf  =  810  Ibs/sq.in. 
V0  =662  sq.in. 

Vf  =  final  air  volume  =  VQ-Abh=662-3.  96x45=  662  -178 
Vf  =  484  cu.in. 


671 


Pa  »  average  drop  of  pressure  across  floating  pis- 
ton =  Ct+Ca(Pal+Paf)0.5 

540+810        540+810 
*  .428  x  — +  ,558x 

=  (.428+. 668)675=. 986x675 
Pa  =  665  Ibs. 

W-  =  30  Ibs. 

V 


TL  *  tension  on  the  rod  =Ks+»  ^ 

30  r 
=  188000+126Qx. 9845+13. 45x24000*  

1260 
=  18800+1240+7700 

TL  »  27750  Ibs. 


Ffflax  =  assume  fibre  stress  1/2  elastic  limit 
=  32500  Ibs/sq.in  =  Fm&x 


27750 
A_  »  area  of  the  recoil  rod  =   •    »  .853  sq.in.=a_ 

32500 

dr  «  diameter  of  recoil  rod  1.127/.853  =  1.04in»  dr 

A'*  area  of  the  recoil  cylinder  =  3.96+1.04  = 
500  sq.in.=  A' 


Dr  *  1.127/5 


2.52in.=D 


r 


D  =1.127/11.30  =3.78K=Da  diameter  of  air  cylinder 
W*  -  30  Ibs. 

STRENGTH  OF  CYLINDERS; 

Test  pressure  =  2  x  service  pressure 
Pt  =  max.  allowaole  fibre  stress  for  cylinders  = 

3/8  elastic  limit  =  3/8  x  60,000  =  22500  Ibs/ 


Raft  »  min.  outside  diameter  of  recuperator 


lao 


1.S9  A 


22500+810 


Pt-P...  22800+810          21700 

U    <a  1 


=  1.958 
Dao  =  3.92  in. 


672 


ro 


ro 


/ 
/ 


22500+4750 


82500-47S0 


•1.561 

•  3.122in. 


W  *  «in.    width   between   adjacent   cylinders 


PaaxDr*Pafxpa 
1.5Pt 

11950+3060 

33750 
.445 


4750x2.52+810x3.78 
1.5x22500 

.445in. 


MAXIMUM  AND  MINIMUM  THROTTLIH9  AREA. 

75   ID/ID     Gun  105m/m 

Normal      Super        How. 

maximum    throttling   area 

(at   0|  elev. ) 


.098x3.96 


673 
.7727, 


.772x21.7  16.72 

*  1 —    »  rr-rr  .422 

(3560-1980)*  39.75 

.772x35.  27 

(6180-1980)  64-81 

.772x47.4  36.60 

'   1     =  .374 

(11550-1980)* 

tfh   »   .422  sq.in. 

fg   *  minimum   throttling   area    (at   80° 
elev. ) 

•772  vr  16.72  16.72 


(7044-1980)*  71.16 

.237  sq.in. 
27  27 

*  =  IS27F  -221  s*iD' 


(16900-1980 )s 

36.6          36.60 


282   sq.in 


129.7          (18800-1980) 

.221*. 282 
a   ________    =    ^25    sq.in.  =    ws 

2 

LAYOUT  0?  RECUPERATOR  FORGIHQ 

If  =  overall  length  of  forging  =  1.5  *  bQ  »  1.5  *45  * 

67.5  in. 
DP  -  diameter  of  recoil  cylinder  2.52  in. 


674 


Da  =  diameter  of  air  cylinder  3.78in, 

la  =  length  of  air  column  58. 5  in. 

Wr  *  area  of  connecting  channel  between  re- 
coil and  recuperator  cylinder. 

Wa  =  3.5  to  4.3  Wh 

=  4x.422  = 
Wfl  =  1.70  sq.in.     da  =  1.468  in. 

D1  =  maximum  depth  of  recuperator  cylinder 
below  recoil  cylinder  = 


=  2.52  -.734 
D'  =  1.786in.' 

Wc  =  const,  channel  area  from  regulator 

valve  =  4.3  Wh  =  4.3x.422 
Wc  =  1.814 

ho  =  depth  of  const,  channel  area,  Wc  in  inches. 
h0  =  0.2Da  =  .2x3.78  =  .756in.=b0 

a  =  extreme  area  of  regulator  valve  = 
W?    73.5x.~422 


73.5 


D|    (3.75)* 

a   *    .933   sq.in.  y 

da   =   diameter   of   entrance   channel   -  9.675   -—  = 

422 
0.675   =    1.09in.-^da 

3.75         

DESIGH     OP     REGULATOR. 

a  =   area  at  base   of   regulator   valve   =   73.5  — —  = 

73.5x.422 
.93   sq.in   =    a 

(3.75)2 

h  422 

d.  =   dianeter   of   a.    9.875  —  =   9.675 =   1.09=da 

Da  3.75         i 

da  *  diameter   of    upper   and   lower   valve   sten  =  0.6da 
.6x1.09   =0.655   =  .d_ 


675 


at  =  cross  section  of  stem  =  0.36a=.36  *.93  = 

.335sq.in.  -a. 


^,  =  diameter  of  inside  foot  of  valve  stem  = 

1   .5  to  .6  da 
ai 

=  .55*. 655  =  .36in.  =  da 


MM. 


c1  =    length   of    flap   at  base   of   scale  =    1.571  da  -.4725 

1.571    x    1.09  -.4725    -      1.237  in. 
c  =    effective   circumference  =    1.571   *    1.09   +   .4725 

2.184in.=c 
Rt  =   reaction   on   tbe   valve   at    short    recoil 

3    +  500*.  335 


=  3950+170 
Rt  =  4120 

SPIRAL  SPRIHG  DESIGN  FOR  REGULATOR  VALVE. 

fs  =  maximum  torsicnal  fiber  stress   120,000  Ibs/sq.in. 
U  =  torsional  modulus  10,500,000  Ibs/sq.in. 

ds  =  let  dianeter  of  the  spiral  be  4  diameter  of 
the  wire 

8PD  32P       10.16P 

*§  =  r>   d*  -  —  --T— 


676 


£P"         496    ~ 
«  3.19  A  =  3.19( )°  =*  3.19  x  .0643 


ds 

f        120000 


.205in. 


D8  *  diameter  of  helical  spring  4  x  d  »  D_  »  .82in. 


deflection  per  coil 
nfsDI 


Gdg       ia500.000x.205 
f  =  .118 

496 

-  =  4200  Ibs.  per  inch  of  deflection  required 

.118 

248 

-  =  1162  spring  const. 

.213 

4200 

-  »  3.61  effective  coils 

1162 

n  =  no.  of  coils  =  3.61  +  1  =  4.61   use  4.5  coils 

Ph 
Rs  =  load  on  spiral  springs  at  0°  elev.  =  -—  -  Pa 

*l 

o  c  c  r\ 

=  (  --  500).  93  =  400  x  .93 
3.96 

Fs  =  372.  Ibs. 

Wp  ^422 

h"  lift  of  valve  at  long  recoil  =  —  -  -  =  .193 

c   2184 

h"  *  .213  inches       Valve  seat  clearance  =  .02 


"W      25  .213in> 

h"  -  lift  at  short  recoil  =  —  »  «  .1144 

c    2 . 184 
h"  =.1144Jn. 

Rsc  =  load  at  solid  height  of  apiral  spring  -j-  x  373  » 

496  Ibs. 

hgc  =  deflection  from  free  to  solid  height  of 
spiral  =  2h"  =  2x.213 

hsc  =  .426in. 


677 


g 

Sg*  spiral   spring   const.   — —  =  »  1162    #   *  S_ 

h__        426  ~_ 

O  C  *-^ -.^ii^^MiP-^Bta- 


g  *  spiral  spring  reaction  at  sbort  recoil 

Ss(h"  +  .02)»1140(.1144  +  .02) 
R'  -  153  Ibs. 


INITIAL 
DEFLECTION 


WORKING 


*« 

N 
<0 


DEFLECTION 


» 


SOLID 
HEIGHT 


.1OG9 


-.2.13 


« — .»O65 


*  load  on  Belleville  at  sbort  recoil 


18800 
(  -    .500).  93-.  355x500 

3.96 
(4750-500).  93+177.  5 


3770    # 


at   solid   beigbt   of    Belleville   washers 

'   I      Rb   =    !   x   3770 
Rbo  =   5650* 

hu0  *  deflection   of   Belleville   from  free   to  solid 

height   3h'=3x.H44 
hbo  «    .343* 


678 


Sb  *  Belleville  spring  const 
Sb  -  16080 


6512 
.343 


16080 


*o 

h- 

Si 


CU 

10 


-.ie — • 


*-.1064- 


-.646* 


DESIGN  OF  CAM  MECHAHISM  AND  LAYOUT. 

g  =  ratio  of  can  movement  to  valve  movenent  usually  5 

X  =  distance  valve  should  lift  to  engage  Bellevilles 

S^  =  working  deflection 

hg  -  initial  corap.of  spiral  spring 

h0  =  clearance  of  valve 

h  =  lift  of  valve 

hjj  =  initial  compression  of  the  Bellevilles. 

X  =  —  {ss(bs+h0*h)+Sb(bb-i-h0+h)-[(P-Pai)a+Paial][' 


Rs=S8hs+Ss(h+li0)    (Steins   of    two   springs    are   in   con- 
tact) 


(P-Pai). 93+500*. 355 


1162*. 1066+11. 62 (h+. 02) 
+16080*.  1144*16080  (h+.  02) 


679 


124  +1162h+23+1840+16080h+322=(p-pal).  93  +  177. 2 
17242h+2132= (p-pai ) .93 


18500 
.098AVr 

<P-Pal)« 

.098x3.96*7, 


2.184(P-Pai)2 


.178V, 


ai 


(1) 
. 098x3. 96*V 


ch»2.184  h 


~   C  ! 


.178 


a 

b 

o 

b 

V 

2290 

Pai>¥ 

2660 

360 

.0194 

51.58 

1.00 

5.  60 

3000 

7oo 

.0378 

54.77 

2.07 

11.62 

3250 

950 

.  0512 

57.01 

2.92 

16.  40 

3500 

1200 

.0648 

59.16 

3.83 

21.50 

3750 

1450 

.  0781 

61.  24 

4.  78 

26.  90 

4000 

i7oo 

.0917 

63.25 

5.  8O 

32.  60 

4250 

1950 

.  1050 

65.  19 

6.  85 

38.50 

45OO 

2200 

.  1188 

67.08 

7.95 

44.  6O 

Normal 


LENGTH  Of  RECOIL 

80°  Elevation 

Vr  =  21.70  corresponding  (P~Pai)  from 

curve  3500 
Ps  =  (3500+. 500)3. 96=4000x3. 96-15820=Ps 


680 


-226+ . 232x4000-226+930 
RBajt-1160  Ibt. 

Kg+»rsin0 
Ps   *       2nd8 R«« 

l-2nr 

Ka+1230x.9848 

15820  -  — 1160 

1+.0388 


16450-K.+1210-1200  N  y. 
Kg  «  16440  -  1.05    [    ' 


2fi  uVf' 

b.+ (.096+. 0003d) 

v 


,,.        1.05x10640       11200 
16440  »  -  


b+.167  b+.167 

16440b+2750-11200 

8450 
b  »  »  .513 

16440 
b  -  6.17ia. 


Super 

Vr  »  35  corresponding  (p~Pai  )*4100  Ibs 
Pg  -  (4100-500)x3.96=4600x3,96 
P,  -  182001 

K8+Wrsin0 


Ls 


2nd, 
1  + 

l-2nr 


226+.  232x4600-226+106 


Rmax  '   130° 


Ka+1210 

P.  »  18200  =  -  -  1300 
1.0388 

18880»KS+1210-1350 


681 


W_V| 
K,  «  18020  lb«.  «  1.05  I— - 


1.05x27600 

X   -^— — ^— — — 

b+.145 
19020b+2760»29000 


b  «  16.55in. 
Howitzer 


Vf»47.30  ft/s.c.   (P-Pai)-4620  Ibt 
P3  =  5120  "  3.96  =  20250  Ibs.  « 
Ks+Wrsin0 


2ndb 
1-  2nr 


~  Rmax 


Raax  s  Ct't'CtP  *  226+.  232*5120  »  226+1190 


Rmax  '  142° 

IT8-1240 
Pg  »  20250  -  --  1420  -2100OK.-1240-1470 

1.0388 

Kg»21230  lbs.= 

21230  =  i- 

b+.267 
21230b  +5630=54500 


b'-        -2.3  feet 
2123 

b"»27.6  inches. 


bs  =       (80-0)+30 
60 


1(80-0)  +30 


682 


1(80-0) +30 

4 

1.05x51900 
b+.267 

655000 


54500 


-  +.267 
12 


655000 
b"+3.21 


b"+3.21 
C 


— 

3.96 


Ks+Wrsin0 


2nd 


-  C. 


1-2  nr 
Kg+1260  sintf 


-  226 


Pc   = 


K_  +  1260sin£) 

9 

1.098 
,178  Vr 


1.0388 
-  212. 

.178x47.30 


8.43 


.178 


0° 

20 

25 

35 

50 

65 

75 

80 

b  " 

45 

43.75 

41.  25 

37.5 

33.75 

31.25 

30 

*, 

13600 

13950 

14710 

16050 

17720 

19000 

19700 

s  inflf 

.  3420 

.  4226 

.  5736 

.  7660 

.9063 

.9659 

.  9848 

K.* 

»r 

•  in^ 

14030 

14480 

15430 

1702O 

18860 

20220 

20940 

Km*r  »int 

r 

12800 

15200 

14100 

15500 

17200 

18900 

19100 

1.098 

P. 

12600 

13000 

13900 

15300 

17000 

18700 

18900 

(P.-P.i) 

1060O 

11000 

11900 

13300 

15000 

i67oo 

16900 

P^-KV 
3.96 

2680 

2*780 

3000 

3360 

3790 

4220 

4270 

683 


(p-pai)¥ 

51.77 

52.  73 

54.77 

57.97 

61.56 

64.96 

65.35 

h 

.  1628 

.  1596 

.  1536 

.  1452 

.  1363 

.  1297 

.  1286 

1162(.1065+.02+h)-H6080(.1144  +  . 


.93+500X.355] 
1 


Ps-1980 
(1162h+147+16080b+2150l  ( )    .93+178] 


16080  3.96 

— (17240h+2300-.235Ps-290)=— - — (17240h-235Ps 

16080  16080 


+2010 
1.072h-(.00001463P8-.1251> 


16 

h 

1.072h 

PS 

1.463 
105   8 

b 

X 

5X 

20° 

.1628 

.1749 

12600 

.1842 

.0591 

.1158 

.580 

85° 

.596 

.1715 

13000 

.1900 

.0649 

.1060 

.530 

35° 

.1536 

.1648 

13900 

.2034 

.0783 

.0870 

.435 

50° 

.1452 

.1559 

15300 

.2390 

.1139 

.0420 

.210 

65° 

.1368 

.1469 

17000 

.2490 

.1264 

.0205 

.103 

75° 

.1297 

.1392 

18700 

.2735 

.1484 

.0008 

.004 

80° 

.1286 

.1380 

18900 

.2765 

.1514 

.0000 

.000 

Counter  Recoil 
Buffer,    constant   orifice   and   port   Areas. 

R'    =   packing  friction  at    end   of   counter   recoil 
Ct   =   226  Ibs. 

d     »   recoil   length  during  buffer   action  T"" 


684 


0.238Wr 


.238x1260                       300  300 

z  —  —  —  ^—  —  —  —   =        i  a  — 

15^1260+226     189+226  415 
db  -  .723  ft  «  8.7  in. 


A       db 
d^  «  length  of  Duffer  rod  *  1.2  x  dfax  —  ,  1.3 — 

Aa  r 

1.2  x    .723 

,  »   .3  ft  »  db  -  3.6  in. 

2.9 
Length  of  buffer  chamber  *  1.2   to  1.3  db' 

dg  »  4.5  in. 


allowable  counter  recoil  velocity 

2.5  to  3.5  ft/sec. 

total  counter  recoil  friction  -  aax 
elev. 

Wr(»in  0t0.3  cos  0a 
Ct  +  CtE  -  -  ]+0.3Wr 


3.96 
.1736 


..3X1260 


«  226  +.232(.2615  )+66 

Rt  "  290  Ibs.  Max.  elevation 


,  +C  ( - )  +.3HP«  226+96+378 

A 

'£  *  700  Ibs.   Win.  elevation. 

pai+pap 


max.  recuperator  pressure  =  P^ 

500+750  2 

— »  625 


685 


c'recoil  orifice  at  80°  elevation 

KAX 

•        (K»1.25) 


1.25«3.96  x  2.5  24.6  24.6 

— — — ^^— — -— ^— — — —^^— ^^^— ^^^— -—   =      i  3  — — — 

13 . 2/625x3 . 96-290-1260x9848  13.2/670    411 
.06  sq.in 


0i 


34.5 34.5 

'o  "  = 


13.2/PaA-R{    13.2/2470-700   13.2x38.35 

34.5 
506 


*  .0682  sq.in. 

—  =  .0641 

KAV 


entrance  buffer  area  = 


13.2       0.00894A«V« 


1.33x3.96x3.5 


13.2  .00894x3.96  x3.5 

625  - 


(.0641)2 


/ 1        1.395 

1.395  / *       »  .097  sq.in. 

625-418 

Lay  out  entrance  area  of  buffer,  with  required 
depth  of  groove,  decrease  depth  of  groove  to 
zero  at  end  of  buffer  dJ.. 


CHAPTER   X. 

RAILWAY  GUN  CARRIAGES. 

TYPES  OF  MOUNTS.     For  coast  defense  or  other 
use  of  heavy  artillery,  it  has 
been  accepted  that  mobility  is 
of  great  importance. 

Materiel  in  permanent  emplacements  is  more  readily 
subjected  to  attack.   Further  with  long  coast  lines 
it  is  impracticable  to  supply  enough  permanent 
batteries  for  adequate  protection.   By  introducing 
heavy  mobile  artillery  ire  increase  the  protection 
and  develop  the  advantage  of  concentrated  fire  at 
any  one  point  when  needed. 

Railway  artillery  meets  the  demand  for 
mobility  in  a  very  satisfactory  degree.   Very 
heavy  weights,  as  occur  with  large  caliber  guns 
and  their  corresponding  mounts,  are  most  readily 
transported  by  rail.   Hence  there  has  been  a 
tendency  of  development  along  two  lines;  first, 
a  mobile  railway  carriage  that  is  entirely  self 
contained  and  fired  directly  from  the  rail  and 
(2)  a  mobile  mount,  transported  by  rail  but  set  up 
on  a  semi-fixed  emplacement.   For  extreme  Mobility 
the  first  is  most  useful,  wherein  for  coast  defense 
work  the  second  plan  offers  many  advantages. 

Railway  carriages  have  been  developed  along 
the  following  lines.  In  their  methods  of  firing. 

(1)  Sliding  carriage  type  with  no 
recoil  mechanism  ,  the  carriage 
merely  sliding  back  during  the  re- 
coil along  special  constructed  rails 
or  guides,  trucks  being  disengaged. 

(2)  Railway  carriages  with  a  recoil 
system,  the  whole  carriage  in  ad- 
dition recoiling  on  special  ways  on 

887 


688 


rails,  the  trucks  being  disengaged, 
or  the  trucks  being  engaged  and 
the  secondary  recoil  being  direct- 
ly along  the  rails. 

(3)     Fixed  or  platform  mounts.   With 
light  railway  artillery,  the  car 
is  held  stationary  by  suitable  out- 
riggers and  we  have  usually  a  bar- 
bette type  of  mount,  mounted  on 
the  car.  With  heavier  types,  the 
girder  which  supports  the  tipping 
parts  is  placed  on  a  large  pintle 
bearing  with  sometimes  additional 
support  at  the  tail  of  the  girder 
with  a  circular  way  or  track  for 
all  round  or  sufficient  traverse. 
In  this  latter  type  the  trucks 
must  be  disengaged  and  the  main 
girder  run  on  to  the  permament 
emplacement. 

The  sliding  carriage  type  (1),  was  developed 
successfully  in  France  and  was  considered  satisfactory 
during  the  late  war.   This  mount,  however,  is  sub- 
jected to  the  direct  firing  stresses  with  consequent 
requirements  for  a  very  heavy  girder  and  trunnion 
support.    It  has  on  the  other  hand  the  advantage 
of  doing  awaj  with  a  recoil  system.   At  best, how- 
ever, it  can  be  regarded  merely  as  an  emergency 
type  of  carriage  that  might  be  developed  under 
great  stress  of  war  pressure  and  not  suitable  for 

use  against  moving  targets. 

In  railway  carriages  of  type  (2),  we  have 

virtually  a  double  recoil  systew.   However,  since 
the  recoil  is  designed  for  stationary  service 
as  well,  or  for  the  condition  at  max.  elevation 
where  the  secondary  recoil  is  small,  the  maximum 
reactions  at  the  beginning  of  the  recoil  are  the 
same  as  in  a.  stationary  mount,  with  a  single 
constant  recoil.   When  the  trucks  are  disengaged 
a  specially  built  track  must  be  laid,  and  the 


689 


girder  slides  back  on  friction  shoes,  which  are 
lowered  to  engage  with  the  track.   Mounts  of  this 
type  are  illustrated  in  our  14"  railway  mount 
ME.   When  the  trucks  are  not  disengaged  and  the 
secondary  recoil  takes  place  on  the  track,  the 
bearing  reactions  of  the  truck  wheels  must  be 
suitably  designed  to  sustain  the  additional  firing 
load  and  the  trucks  must  be  suitable  braked  to 
resist  the  secondary  recoil,  and  bring  the  mount 
to  rest  after  the  firing.    When  a  built  up  track, 
trucks  disengaged,  is  used  the  successive  firings 
must  necessarily  take  place  along  the  tangent  of 
the  track,  whereas  firing  directly  from  the  rails, 
permits  the  use  of  a  curved  or  Y  track,  and  con- 
siderable traversing  is  thus  possible  by  the 
firing  taking  place  at  different  points  on  the 
curved  track.   With  railway  carriages  of  type  (2) 
very  little  traversing  is  possible  on  the  mount 
itself  and  therefore  the  track  must  be  laid  very 
closely  in  the  direction  of  firing.    In  railway 
carriages  of  type  (2),  we  are  greatly  limited  by 
road  clearance.   For  clearance,  the  trunnions 
must  therefore  be  in  the  traveling  position  in 
a  low  position.   On  firing  however,  at  maximum 
elevation,  the  recoil  becomes  limited.   To  pro- 
vide for  a  suitable  recoil  at  maximum  elevation 
the  trunnions  are  raised  and  a  balancing  gear 
throwing  the  trunnions  to  the  rear  may  also  be 
introduced. 

With  fixed  or  platform  mounts  of  type  (3), 
the  special  features  are  the  methods  of 
erection  on  to  a  serai  permanent  emplacement 
and  the  disengagement  from  the  traveling  con- 
dition of  the  mount  .  We  may  have  a  center  turn 
table  which  serves  for  the  pintle  in  traversing 

and  the  tail  of  the  girder  is  supported  by  a  suit- 
able circular  guide  which  balances  the  overturn- 
ing moment  and  thus  releases  the  otherwise  bend- 
ing or  overturning  moment  on  the  pintle  bearing. 


690 


With  this  type  of  mount  large  traversing  is  complete- 
ly possible. 

SPECIAL  PIATURS3  IN  THB  DESIGN. 

Recoil  System: 

(1)  The  recoil  should  be  simple  and 
rugged. 

(2)  A  constant  recoil  or  approximate- 
ly constant  for  all  elevations 

should  be  used. 

(3)  A  constant  resistance  to  recoil 
is  satisfactory  since  questions  of 
stability  are  not  usually  of  prime 
consideration,  and  the  recoil  is 
thus  simplified. 

(4)  The  counter  recoil  should  be 
sirople,  an  ordinary  spear  buffer 
being  usually  satisfactory  although 
other  control  may  be  sometimes 
necessary.   Bere  again  counter  re- 
coil stability  is  no  longer  a  con- 
sideration and  high  velocities  of 
counter  recoil  are  not  objectionable 
provided  there  is  no  shock  at  end 

of  counter  recoil. 

(5)  With  very  large  guns  used  at  high 
elevations,  high  pressure  pneumatic 
recuperator  systems  should  be  used 
in  place  of  spring  columns,  since 
the  weight  and  bulk  of  springs  be- 
come excessive. 

(6)  Sleeve  guides  for  the  gun  have 
been  found  most  suitable  and  tne 
various  pulls  should  be  so  far  as 
possible  symmetrically  spaced  about 
the  axis  of  the  bore,  thus  reducing 
tne  bearing  reactions  in  the  sleeve 
and  making  it  also  possible  to  keep 


691 


the  center  of  gravity  of  the  recoil- 
ing parts  close  to  the  axis  of  the 
bore. 

Tipping  parts: 

(1)  The  cradle  should  be  of  the 
sleeve  type  thus  reducing  the 
bearing  pressures  over  guides 
and  clips. 

(2)  The  recoil  and  recuperator  can 
be  strapped  on  with  suitable 
shoulders  for  bearing  surface  to 
take  up  the  recoil  load  from  the 
cyli  nders . 

(3)  The  trunnions  should  be  located 
near  line  through  the  center  of 
gravity  of  the  recoiling  parts  and 
parallel  to  the  axis  of  tue  bore. 
This  reduces  the  elevating  re- 
action during  the  pure  recoil  to 
merely  that  due  to  the  moment  ef- 
fect of  the  recoiling  parts  out  of 
battery. 

(4)  Great  effort  should  be  made  to 

locate  the  center  of  gravity  of  the 
recoiling  parts  as  near  the  axis 
of  the  bore  as  possible  either  by 
symmetrically  distributing  the  re- 
coil rods  and  attachments  or  if 
necessary  introducing  counter 
balancing1  weights.   Thus  the 
whipping  action  during  the  powder 
period  is  reduced  with  a  correspond- 
ing reduction  in  the  elevating  arc 
reaction  during  the  powder  period. 

(5)  With  high  angle  fire  ^uns  or 
howitzers,  the  trunnions  may  be 
thrown  to  the  rear,  and  balancing 


692 


gear  introduced,  thus  making  long 
recoil  possible.   Another  plan  for 
accomplishing  the  same  results  is 
to  raise  the  trunnions  before  fir- 
ing. 

(6)  The  trunnion  bearings  should  be 
supported  on  springs  during  travel- 
ing, though  compressed  so  we  have 
solid  contact  during  firing. 

(7)  To  reduce  the  friction  during 

the  elevating  process,  ball  or  roller 
bearings  should  be  introduced  in 
the  trunnion  bearings,  or  in  an 
inner  trunnion  should  be  introduced 
of  smaller  radius  than  the  main 
trunnion  for  reducing  friction  on 
rotating  the  tipping  parts. 

LIMITATIONS  IN     With  heavy  artillery  mounts, 
BRAKE  LAYOUT,   either  railway  or  lor  permament 

or  mobile  emplacements,  counter  re- 
coil stability  is  not  a  consideration. 
On  the  other  hand  we  are  limited  to 
a  maximum  allowable  buffer  pressure  in  the  counter 
recoil.   With  counter  recoil  systems  which  come 
into  action  towards  the  end  of  counter  recoil, 
practically  the  entire  potential  energy  of  the 
recuperator  most  be  dissipated  by  the  buffer  over 
a  relatively  short  displacement.   Now  since  the 
potential  energy  of  the  recuperator  is  a  con- 
siderable fraction  of  the  energy  of  recoil,  we 
see  that  the  buffer  reaction  is  of  a  magnitude 
comparative   with  the  brake  resistance  during  the 
recoil.   Further  the  effective  area  of  the  c'recoil 
buffer,  due  to  constructive  limitations,  is  necessarily 
considerably  smaller  than  the  effective  area  of   the 
recoil  brake.   Hence  the  buffer  pressures  with  a 
short  c'recoil  buffer,  become  very  great.   This  is 
especially  pronounced  with  a  short  buffer  and  high 


693 


angle  fire  gun  where  the  unbalanced  recuperator 
energy  is  necessarily  great,  when  the  gun  c'recoils 
at  lovi  elevations.   As  to  the  limiting  allowable 
buffer  pressures,  no  hard  and  fast  rule  can  be 
made,  but  it  is  certain  that  the  buffer  pressures 
in  many  of  our  mounts  are  rather  too  high  for  light 
construction,  requiring  heavy  and  strong  buffer 
chambers . 

With  recoil  brakes  having  a  continuous  rod 
extending  through  both  ends  of  the  cylinder,  the 
effective  area  of  the  buffer  must  be  necessarily 
very  snail  and  the  stroke  of  the  buffer  short  due 
to  the  fact  that  during  the  recoil  it  is  important 
that  the  void  displacement  be  not  too  great. 
Hence  this  type  of  brake  with  continuous  rod  and 
enlargement  for  c 'recoil  buffer  ram,  has  inherent- 
ly excessive  buffer  pressures.   It  is  very  important 
with  such  mounts  to  maintain  a  minimum  recuperator 
energy,  that  is  to  use  the  minimum  recuperator  re- 
action  combined  with  a  low  ratio  of  compression, 
consistent  with  proper  c'rscoil  at  maxiuum  elevation, 

To  reduce  the  -buffer  pressure,  the  c'rscoil 
regulator  should  be  effective  throughout  the  recoil, 
and  thd  effective  area  of  the  buffer  should  be  as 
large  as  possible.   This  actually  has  been  obtained 
constructively  in  our  16  inch  railway  mount,  the 
buffer  area  being  equal  to  that  of  the  recoil  brake 
and  c 'recoil  regulation  taking  place  throughout  the 
rscoil.   The  buffer  pressures  are  therefore  compar- 
able with  the  brake  pressures  during  recoil. 
DESIGN  LAYOUT  OF     Assuming  a  preliminary  layout 
RECOIL  SYSTEMS.    has  been  made,  the  weight  and   the 
ballistics  of  the  gun  given,  we 
may  estimate  from  previous  mounts, 
the  probable  weight  of  the  recoil- 
ing parts  and  tipping  parts. 

Therefore,  we  will  assume  the  following  data 
given  or  estimated  from  previous  mounts: 


694 


Wr  *  neigh!  of  recoiling  parts  (estimated)  (Ibs) 

d  *  diameter  of  bore  (in) 

v  »  nuzzle  velocity  (ft/»ec) 

w  *  weight  of  projectile  (Ibs) 

•  »  weight  of  charge  (Ibs) 

pbm  3  maximum  powder  pressure  (Ibs/sq.in) 

b  =  mean  length  of  recoil  (ft) 

0m  »  maximum  angle  of  elevation 

0£  *  minimum  angle  of  elevation 

u  «  travel  up  the  bore  of  the  projectile  (ft) 

Calculation  of  E  and  T: 

From  the  principle  of  Interior  Ballistics, 
we  have,      R 

PU,  =  -  d"  pb|n  =  max.  total  pressure  on 
breech   (Ibs) 


wv1 
Pe  »  —  -  »  average  force  on  projectile 

(Ibs)    _ 

e  -  u[(~  ^  -  1)±  /<l  -  |Z  -i)«  -  1  ]  .  twice  the 
P«  P«          travel  of 

projectile  to 
max.  pressure  in 


27  .    u 

Pftw  *  r—  e   .      P_  *  total  pressure  on  breech 
4     (e+u)3  .         . 

when  shot  leaves  muzzle. 


wv  +  4700ii 

'   »  max.  velocity  of  free  recoil  (ft/sec) 


v 

»  velocity  of  free  recoil  when  shot 


"r 

T0  •  -  -  »  time  of  travel  of  shot  to  muzzle  (sec) 

2(Vf-Vfo)   Wr 

Tt  *   -       -T  •  time  of  free  expansion  of  gears 
"ob          t   . 
(sec) 


695 


T  »  t0+t  =  total  powder  period  (sec) 
*o 

.w+0.5  w 

xfo  *  '  -  JJ  --  'u  *  f  ree  displacement  when  shot 
leaves  muzzle  (ft) 


+  vfo(T~to)*  free  displacement 
*r  of  recoil  during 

free  expansion 
of  gas  (ft) 
X*0+X*»0  *  free  recoii  during  powder  period,  ft. 


Resistance  to  Recoil 

Knowing  &  and  T  we  may  immediately  calculate 
the  total  resistance  to  recoil  for  any  elevation, 

from  the  formula:-      i 

I  "r7! 
K  ,  '•   •  (Ibs) 

b-B+VfT 

With  spear  buffers  effective  daring  the  latter 
part  of  counter  recoil,  in  order  to  reduce  the  buffer 
pressure  (Ibs/sq.in)  the  effective  area  of  the 
spear  buffer  is  made  greater  than  the  area  of  the 
recoil  rod.    Now  due  to  the  relatively  snail  area 
of  the  c'recoil  throttling  areas,  the  sudden  with- 
drawal of  the  plunger  of  the  c'recoil  buffer  on 
firing,  prevents  a  ready  flow  of  oil  into  the  space 
vacated  by  it.    Hence   we  would  have  a  very  great 
resistance  set.  up  unless  a  by-pass  or  void  is  in- 
troduced.  Due  to  difficulty  in  obtaining  a  suf- 
ficiently large  by-pass  together  with  additional 
constructive  difficulties,  it  is  customary  to 
partially  fill  the  recoil  brake  cylinder  leaving 
a  void  in  the  cylinder. 

To  calculate  the  void  displacement,  fig.  (1) 
let 

A  «  effective  area  of  recoil  piston  (sq.ft.) 

A1  ^effective  area  of  recoil  piston  on  c'recoil 
plunger  side  (sq.ft) 


696 


697 


db  =  length  of  buffer  or  plunger  (ft) 

S  =  length  of  void  displacement  (ft) 

tg  *  time  of  recoil  through  the  void  (sec) 

then,  we  have  A(db-S)-A'd|j  *  0 

hence     U~A  '  )db 

S  =  -  (ft) 


The  resistance  to  recoil,  with  a  void,  becomes 
7  mrv! 
b-E+Vf  (T-ts) 

To  compute  ts  we  proceed  as  follows: 

(1)     If  the  void  displacement  is  less 
than,     (w+  ![  )u 

s   -      at) 


then  u1  = 


wrs 


(ft) 


t3  =  -  (2.3  log  —  +  —  +  2  )  (sec) 
a          e    e 

(e+u)v 


e  is  obtained  from  the  previous  inertia  ballistic 
calculations. 

(2)     When  the  void  displacement  is 
given  byL 


then 


where 


(o  - 


-tn) 


6mr(Vf-Vfo) 

wv+4700  w 


698 


3  U 

«"  V 


Since  the  above  expression  is  a  cubic  equation 
in  tg,  we   may  more  conveniently  solve  it  by  sub- 
stituting trial  values  for  ts  until  we  obtain  the 
approx.  value  of  s. 

In  an  approximate  design  if  we  assume  no  void, 
K  may  be  calculated  immediately  without  the  com- 
putation of  E  and  T,  from  the  formula: 

v  *  *  m  VS         *  W4700+WV 

K  »  i  mrvf  »     —xhere  Vf  »— — — 

[b+(. 096*. 0003d) — L  ]  W_ 

v 

d  *  diam.of  bore  (in) 

v  »  muzzle  velocity  of  projectile  (ft/sec) 

b  »  length  of  recoil  (ft) 

u  »  travel  of  shot  up  bore  (in) 

w  =  weight  of  shell 

w  *  weight  of  charge 

or  and  Wr  -  mass  and  weight  of  recoiling1  parts. 

Estimation  of  Pullsr-Recuperator  and  Brake. 

Cylinder  preliminary  layout. 
If 


Rg  =  guide  friction  (Ibs) 

0m  *  max.  angle  of  elevation 

0£  *  initial  angle  of  elevation 

Rp  *  total  packing  friction  (Ibs) 

B  =  total  braking  resistance  (Ibs) 

Pb  »  brake  cylinder  pull  (Ibs) 

Fvi  *  initial  recuperator  reaction  (Ibs) 

m  *  ratio  of  compression  (assumed  from  1.3 

to  1.7) 

Fyf  *  m  Fvi  =  final  recuperator  reaction  (Ibs) 
1  =  length  of  cradle  and  gun  sleeve  (in) 
eb  »  distance  from  center  of  gravity  of  re- 
coiling parts  to  center  of  pulls 


699 


n  *  coefficient  of  guide  friction  »  0.15  ap- 

prox. 

x1ana  x2  =  coordinates  of  front  and  rear  clip 
reactions  from  center  of  gravity 
of  recoiling  parts  (in) 

A  *  effective  area  of  recoil  brake  piston 
Ay  *  effective  area  of  recuperator  piston 
ar  *  area  of  recoil  brake  piston  rod 
av  s  area  of  recuperator  piston  rod 

Then  K  a  B+Rp+Rg-Wrsin0.    As  a  first  approximation, 
we  will  neglect  R  and  assume,  R-  »  n  Wr  cos  0, 
then  B  =  K+Wr(sin0-n  cos  00  (Ibs).  For  the  initial 
recuperator  reaction,  Fyj  =  1.3  Wr(sin  J0+n  cos  0) 
(Ibs)  and  since  B  =  Ph+Fvi»  tn®  total  braking  (Ibs) 
ire  have  for  the  initial  hydraulic  pull, 
Ph  «  K-Wr(0.3  sin0+2.3  n  cos  0)  (Ibs).    In  a 
preliminary  design,  the  following  are  working 
pressures,  consistent  with  the  packings:- 
Ph  max  *  350°  to  4500  Ibs/sq.in.  brake  cylinder- 
pvi  *  1000  to  1500  Ibs/sq.in.  recuperator  cylinder 
Further  let  fm  =  max.  allowable  fibre  stress  in 
the  various  piston  rods.   Hence  for  the  recoil 
brake,  we  have,  for  "n"  cylinders 

1.2  Ph          Ph 
Ar  »  -  ;    A  =  -   (sq.in) 

n  fm         npn  max 

(the  factor  1.2  is  to  allow  for  the  acceleration 
of  the  rod  during  the  powder  period),  and  the 
diam.  of  a  recoil  cylinder,  becomes, 


(in)    and  the  diam.  of  a  brake 


0.785 
The   recuperator   dimension,    for    "n"  cylinders 

becomes,      p^f         Fy. 

a  =     ' 


where  Fvf   =   roFvi   -    1.5   to   1.7  Fvi 


700 


The  dia«.  of  a  recuperator  cylinder,  becomes, 


and  the  diam.  of  the  recuperator 
0.7854    roa 


From  these  dimensions  a  preliminary  layout  of  the 
brake  and  recuperator  cylinders  may  be  made,  and 
the  positions  of  the  center  lines  of  the  various 
pulls  located  with  respect  to  the  axis  of  the 
bord  or  center  of  gravity  of  the  recoiling  parts. 
If  now, 

eh  =  distance  from  center  of  gravity  of  re- 
coiling parts  to  line  up  action  of  hy- 
draulic brake  pull  (in) 

e?  =  distance  from  center  of  gravity  of  re- 
coiling parts  to  line  of  action  of  re- 
cuperator reaction  (in) 

e^  -  distance  from  center  of  gravity  of  re- 
coiling parts  to  line  of  action  of 
resultant  pull  (in) 
then      Fyi  ey+Pneh 

Cw  =  (in)  where  B  =  F  .  +  P 

b  vi    b 


Calculation  of  packing  friction. 

To  estimate  the  packing  friction,  we  must 
assume  the  diameters  of  cylinders  and  rods,  as 
approximated  from  the  previous  calculations: 
then 

Rp  =  2  .05  itd  w  Pmax  where  d  =  diam.  of  the 
various  rods  and  cylinders 

Wp  =  corresponding  width  of  the  packing 
Pmax  =  (Dax'  pressure  in  the  various  cylinders 
The  component  packing  frictions  for  the  re- 
cuperator and  brake  cylinders,  consist  of  the 
stuffing  box  and  piston  frictions  respectively. 

For  the  brake  cylinder, 
Rph  =  I  .05  n(drWr+D  *d)ph  max. 


701 


where 

dr  =  dian.  of  brake  rod 
D  =  iiam.  of  brake  cylinder 
tfr  *  width  of  stuffing  box  packing 
Wjj  *  width  of  piston  packing 
For  the  recuperator  cylinder  R   =  2  .05  «  (dywy 

+Dvwv>Pv  max. 
where  dv  =  diam.  of  recuperator  rod 

Dy  =  diam.  of  recuperator  cylinder 
Wy  =  width  of  stuffing  box  packing 
Wy  =  width  of  piston  packing 

then  Rp  =  Z  Rph  +  ZRpv  =  total  packing  friction 
If  Pn  =  the  total  hydraulic  reaction 

P£  =  the  total  tension  or  poll  in  the  brake 

rods 

FV  =  the  total  recuperator  reaction 
.Fy  =  the  total  tension  or  pull  in  the  re- 
cuperator rods 
then  Pn=Pn+2Rph  ;  Fv=FV2Rpv 

Guide  Friction 


We  may  now  estimate,  more  exactly,  the  guide 
friction.   We  have  two  cases, 

(1)  When  the  resultant  pulls  are 
symmetrically  balanced  around  the 
axis  of  the  bore 

(2)  When  the  resultant  pull  is  off 
set  from  the  axis  of  the  bore. 

In  (1)  we  have  simply  R-  =  n!Tr  cos  Gf  (Ibs) 
In  (2)  we  have 

2n(B+R  )eb+nWrcos0(xi-xa) 

R«  = (Ibs) 

1+2  n  eb 

where  n  =  0.15 

xt  and  xa  are  the  front  and  rear  clip  reaction 
coordinates  with  respect  to  the  center  of  gravity 
of  the  recoiling  parts. 


702 


1  »  distance  between  clip  reactions  and  length 

of  sleeve  in  cradle, 
e^  *  distance  down  from  bore  to  resultant  line  of 

action  of  mean  total  pull  (B+Rp).   In  general, 
however,  we  may  neglect  R_  as  small  compared  with 
8,  and  2  n  eb  as  small  compared  with  1,  then, 

2n(K+Wrsin0)eb+nW_cos0(x  -x  ) 

R  = x   '     (Ibs) 

«  1 

The  term  n  Wr  cos  0(xt-x8)  is  usually  small  com- 
pared with  2n(K+Wrsin0)e^  and  further  very  often 
we  may  assume  xt  =  xz  approx.,  hence, 

2n(K+«_  sin0)eb 
Rg  - (Ibs) 

which  is  usually  sufficiently  accurate  for  ordinary 

calculations . 

It  is  to  be  particularly  noticed,  that  when 

the  pulls  are  offset  from  the  axis  of  the  bore, 
the  guide  friction  increases  on  elevating  which 
is  exactly  opposite  to  the  condition  of  sym- 
metrically and  balance  pulls  about  the  axis  of 
the  bore,  when  Rg  =  nWr  cos  0. 

[nitial  Recuperator  reaction, 


The  required  initial  recuperator  reaction 

is  given  by  the  following  formula: 

n  cos  0   (x.  -x  )  , 


1+2  n  er 


2ev  n 
1  - 


1+2  ner 
•  here  Rpy  =  2  .05  *  (dv«v+DtfWv )pvi 

=  assumed  initial  recuperator  pressure 
n  cos  0m  )«r 


703 


Ay  »  assumed  effective  area  of  recuperator 

piston 

d?  *  dia>.  of  recuperator  rod  (in) 
Dv  =  diam.  of  recuperator  cylinder  (in) 
wy  =  width  of  stuffing  box  packing  (in) 
Wv  =  width  of  piston  packing  (in) 
1  *  length  of  sleeve  or  distance  between 

guide  reactions  (in) 

ev  =  distance  from  center  of  gravity  of  re- 
coiling parts  to  resultant  line  of 
action  of  Fv 

er  =  mean  distance  from  center  of  gravity  of 
recoiling  parts  to  guides  (  =  0,  for 
sleeve  cradles) 

x±  and  xa  =  coordinates  of  front  and  rear 

clip  reactions  from  center  of 
gravity  of  recoiling  parts  in 
battery  (in) 

n  =  coefficient  of  guide  friction  (=0.15) 
0m  =  angle  of  max.  elevation 

The  above  formula  is  complicated  and  the  fol- 
lowing formula  is  usually  sufficiently  accurate 

and  takes  into  consideration  as  well  the  pinching 
action  between  the  guides  and  clips, 


vi  =^ 

W  sin^gj+R 

-)(lbs)  where  k  =  1.1  to  1.2 

i 

«•••— 

1 

when  ev  is  small  as  with  symmetrically  balanced 
recuperator  pulls,  then  Fvj  =  k[ Wr (sin0m+n  coB/)B)+Rp] 

where  k  =  1.1  to  1.2 

If  we  include  Rp  with  n  Wr  cos  0,  we  may  in- 
crease k,  and  we  have  the  elementary  formula  as 
before  used,  Pyi  =  1.3 (Wrsin0B+0.3  cos0m)  (Ibs) 


704 


Counter  Recoil  Buffer  or  Regulator  Design 

Counter  recoil  regulators  may  be  divided  in- 
to two  general  types, 

(1)  Systems  which  are  effective  only 
during  the  latter  part  of  counter 
recoil. 

(2)  Systems  which  fill  themselves 
during  the  recoil  and  are  effective 
throughout  the  counter  recoil. 

In  type(l)  we  have  a  short  spear  buffer  or 
plunger  entering  the  buffer  chamber  towards  the 
end  of  recoil.  Type  (1)  buffers  may  be  further  sub- 
divided into:- 

(a)  Plungers  attached  to  a 
continuous  recoil  rod,  the  re- 
coil rod  passing  through  a  stuffing 
box  at  either  end  of  the  piston. 

(b )  Ordinary  spear  buffers  with- 
out a  continuous  recoil  rod. 

In  the  design  of  a  counter  recoil  system,  we 
are  primarily  limited  to  a  maximum  allowable  buffer 
pressure,  counter  recoil  stability  in  heavy  artil- 
lery being  of  no  great  importance  since  the  stabil- 
ity limit  on  a  counter  recoil  is  usually  as  great 
as  on  recoil.   Since,  however,  a  considerable  part 
of  the  recoil  energy  becomes  at  the  end  of  recoil 
stored  in  the  recuperator,  we  have  this  energy 
absorbed  in  the  counter  recoil,  by  the  counter  re- 
coil regulator  in  a  short  buffer  displacement,  with 
a  consequent  large  total  buffer  reaction.   We  are 
limited  in  tbe  counter  recoil  brake  usually  to  a 
smaller  effective  area  than  in  the  recoil  brake; 
consequently  the  buffer  pressures  become,  due  to 
constructive  limitations,  very  large.   Hence  it 
is  highly  desirable  to  maintain  as  low  a  buffer 
pressure  as  possible. 

With  any  form  of  spear  buffer  of  type  (1), 
to  reduce  the  buffer  pressure,  the  effective  area 


705 


of  the  buffer  plunger  should  be  as  large  as  pos- 
sible and  the  length  of  buffer  as  long  as  possible. 

In  the  design  of  a  spear  buffer  of  type  (1) 
we  have  the  following  limitations :- 

(1)  The  diameter  of  the  buffer, 
should  not  exceed  a  value,  that 
due  to  the  sudden  withdrawal  of  the 
buffer,  the  void  displacement  in 
the  recoil  brake  should  not  be 
greater  than  the  free  recoil  dis- 
placement during  the  powder  period 
E. 

(2)  The  length  of  the  buffer  should 

not  exceed  a  value  that  during  the 
counter  recoil  before  the  buffer 

enters  its  chamber  the  buffer 
chamber  should  be  completely  filled. 
Let  A  *  effective  area  of  recoil  piston  (sq.ft) 

A'=  effective  area  of  recoil  piston  on  counter 

recoil  plunger  side  (sq.ft) 
Lb=length  of  plunger  or  buffer  (ft) 
Ab  *  effective  area  of  buffer  (sq.ft) 
du  =diam.  of  buffer  chamber 

D  -  diam.  of  recoil  brake  cylinder 
i_  =  diam.  of  recoil  brake  rod 
Now  A  5  0.7854(Da-d»)  ;  A'=0.7854 (D*-dg )  sq.ft. 

Ab=0.7854(dg-d»)   (sq.ft)  type  (l)(a)  buffer, 

Ab=0.7854  dg  (sq.ft)  type  (1)  (b)  buffer. 
Now  for  condition   or  limitation  (1),  we  have 

(ft) 

or        A(Lb-E)       g 

A1  =  A(l-  -—  )sq.ft. 

Lb  Lb 

In  terms  of  the  diameters,  we  have 

E 
D«-dg 


706 


/£         E 
)*  T—  +  d*(l-  —  )   which  gives  us  the 

b   limiting  value  of 

db.   It  is  interesting  to  note  that  when  6=0. 
d]j*dr  or  in  other  words  when  the  diameter  of  the 
buffer  a  plunger  is  made  equal  to  that  of  the  rod, 
no  void  is  required  in  tbe  recoil  cylinder. 

From  the  above  expression,  we  note  that  in- 
c  re  as  ing  the  length  of  the  buffer  decreases  the 

diameter  of  the  buffer  and  thereby  increases  the 
buffer  pressure. 

On  the  other  band  the  c  'recoil  energy  is  ab- 
sorbed over  a  greater  distance  with  a  longer 
buffer,  thus  reducing  the  total  buffer  reaction, 
and  it  is  probable,  that  this  cause  more  than 
effects  the  slight  increase  of  the  buffer  pressure 
dueto  tbe  decrease  of  the  buffer  diameter.   Further 
the  value  of  d|>  is  very  often  entirely  limited  by 
constructive  considerations  alone;  hence  a  long 
buffer  is  highly  desirable. 

In  a  type  (1)  (a)  buffer  due  to  the  relative- 
ly large  value  of  db  required  to  give  a  sufficient 
buffer  area,  the  length  of  the  buffer  depends  en- 
tirely on  the  limitations  (1).   This  type  of 
buffer  will  be  considered  in  detail  later. 

For  the  limitation  (2),  with  a  continuous  rod, 
we  have  a  void  produced  at  the  end  of  recoil  on 
tbe  buffer  side  of  the  recoil  piston.   To  compute 
this  void,  we  have,  with  an  initial  void  in  the 
battery  position  AE,  for  tbe  void  on  the  buffer 
side  of  the  recoil  piston  at  the  end  of  recoil, 
or  the  out  of  battery  position. 

Voidc=Arb-A(b-E=(cu.ft) 

where  Ar  =  area  of  the  recoil  cylinder  (sq.ft) 
Therefore,  Voidc  =  (Ar-A)b  +  AE 

=  arb+AE(cu.ft) 

Now  in  the  c  'recoil,  the  spear  buffer  chamber 
is  evidently  not  filled  until  tbe  void  displace- 

ment has  been  over  run,  and  this  displacement 
a^>  +AE    . 


v 

becomes,   Xa  = 

r 


707 


D* 

AE 
Since  -—  is  small,  for  a  close  approximation,  the 

r  buffer  length  should  not  exceed 
Lb-0.8b(l-  Jt  )=0.8b(l-|f)   ft. 

The  mean  buffer  pressure  may  now  be  computed, 
knowing  the  potential  energy  of  the  recuperator. 

The  potential  energy  of  the  recuperator  is 
given  by  either  of  the  following  expressions: 

pvivo 
**o  *   fv    ^B   ~  D(ft.lbs)  (k=1.3  approx.) 

V 


HO  «  -  (ft.lbs) 

Ar(k-D 

where  Vf=Vo-Avb 

Av  =  effective  area  of  the  recuperator  piston 
(sq.ft) 

Vo«  the  initial  volume  (cu.ft) 

F?i  =  the  initial  recuperator  reaction  (Ibs) 

%i 
m  »  —  •  —  =  the  ratio  of  compression 

Fvi 
Then,  the  mean  buffer  pressure,  becomes 

l»0-(Wrsin0+R0)b 
Pb  »  *r  -  -  -  (Ibs/sq.ft) 


where   Ro=total   packing   and  guide   friction    (Ibs) 
b   =   length   of   recoil    (ft) 

d? 
Lb   =   0.8(1-  ^-)    (ft) 

Ab    =   0.7854   dg    )sq.ft) 


r-T'  (ft'> 


708 


In  type  (1)  (a)  buffer,  where  we  have  a  con- 
tinuous rod  and  enlargement  back  of  the  piston  for 
the  c 'recoil  plunger,  in  order  to  have  a  sufficient 
effective  buffer  area,  the  diameter  of  the  plunger 
must  be  necessarily  large  as  compared  with  a  spear 
buffer.   Therefore,  to  maintain  a  void  displacement 
in  the  recoil  not  exceeding  the  free  recoil  displace- 
ment during  the  powder  pressure  period,  we  must 
have  a  very  short  buffer.   Hence  if 

A  -  effective  area  of  recoil  piston  (sq.ft) 

A'-effective  area  of  recoil  piston  on  c'recoil 
plunger  side  (sq.ft) 

Lb  =  length  of  plunger  or  buffer  (ft) 
effective  area  of  the  buffer 


If  further  d«  =  diam. 


of  recoil  brake  rod 
D  =  dia».  of  recoil  brake  cylinder 
db  =  diam.  of  buffer  chamber 

we  have  db  *  CbD  where  Cb  depends  upon  constructive 

considerations 


[(l-Cg)D«-dr] 
and 

Ab  =  0.7S54(dg-d«)-0.785  (C£  D«-d») 

Now  to  reduce  the  buffer  pressure  it  is  de- 
sirable  to  make  Lb  as  long  as  possible  and  Ab  as 
small  as  possible.   To  do  this  we  must  make  dr  as 
small  as  possible  as  compared  with  D.   This  re- 
quires a  large  effective  area  for  the  recoil  brake. 

Hencs  in  type  (1)  buffer  we  may  reduce  the 
buffer  pressure  by  reducing  the  recoil  brake  pres- 
sure.  If  HO  =  the  total  potential  energy  of  the 
recuperator  we  have 


709 


F    -    V 

(m    k     _   x)    (ft>    lbs)    (k=1.3    approx.) 


=  the  ratio  of  compression 


where    Pyf 


Av  =  effective  area  of  the  recuperator  piston 

(sq.ft) 

VQ  =  initial  volume  of  the  recuperator  (cu.ft) 
then  for  the  mean  buffer  pressure,  ire  have 


W0-(Wrsin0+Rp)b 
AbLb   =   0.785 


Now  (D*-d»)(CgD«-d*) 

- 2 L 


(l-Cg)D*-d« 

If  we  assume  Cb  =  0.7  roughly,  we  have  AbLb=0.785(D2-d* 
hence      Ho-(Wrsin0+Rp  )b 

6   0.785(D*-d*)E 
where 

b  =  length  of  rscoil   (ft) 

E  »  free  displacement  in  the  recoil  during 

powder  period  (ft) 

HQ  =  potential  energy  of  the  recuperator  (ft. 
Ibs) 


RQ  =   total   friction    (Ibs) 


since 


we   have   pb    =    [B0+(WrsinJO+RQ)]   - 

PhE 
where 

Ph  =  total  hydraulic  brake  pull  (Ibs) 

pb  =  assumed  intensity  of  pressure  in  hydraulic 

cylinder  (Ibs/sq.in) 

Therefore,  to  decrease  the  buffer  pressure,  with 
a  type  (1)  (a)  c'recoil  regulator: 

(1)     Lower  the  max.  pressure  in  hy- 
draulic braka  cylinder  during  the 


710 


recoil. 

(2)  Decrease  the  length  of  recoil 

(3)  Decrease  the  potential  energy 
in  the  recuperator. 

We  see  that  the  above  expression  is  fixed  by 
the  free  recoil  displacement  E  during  the  powder 
period. 

BY  PASS  PIPES       In  order  to  loner  the  buffer 
USED  WITH  LARGE  pressure  on  counter  recoil,  when 
SPEAR  BUFFERS.   the  c 'recoil  regulation  is  by  a 
short  spear  buffer  or  plunger, 
it  is  often  necessary  to  in- 
crease the  diameter  of  the  plunger  materially  over 
that  of  the  rod. 

By  the  introduction  of  a  by  pass  and  valve 
(closing  on  the  counter  recoil  Heading  from  the 
buffer  side  of  the  recoil  cylinder  to  the  outer 
end  of  the  void  chamber  of  the  buffer,  the  pressure 
back  of  the  recoil  piston  (on  the  buffer  side)  can 
be  effectively  lowered  without  a  full  void  by 
being  required  in  the  recoil  cylinder  to  take  care 
of  the  sudden  withdrawal  of  the  buffer  plunger 
during  the  first  part  of  the  recoil. 

Let  wa  =  required  area  of  the  by  pass  pipe 

T  *  total  powder  period  (sec) 

tg=tirae  of  travel  through  void  during  the  re- 
coil (sec) 

E  =  recoil  displacement  daring  powder  period 
(ft) 

A  =  effective  area  of  recoil  piston  (sq.ft) 

A'=effective  area  of  recoil  piston  on  plunger 
side  (sq.ft) 

S  *  recoil  displacement  during  void  (ft) 

Lfc  -  length  of  buffer  (ft) 

Pg  =  mean  pressure  in  the  rear  of  the  recoil 

piston  (Ibs/sq.in) 
p1  -  max.  pressure  in  the  rear  of  the  recoil 

piston  (Ibs/sq.in) 


711 


Now  the  total  quantity  of  oil  that  Bust  pass  through 

the  by  pass  pipe,  becomes,  Q  »  A(L^-S)-A'L^  (cu.ft) 
After  the  gun  has  recoiled  the  void  displace- 

ment, the  void  bade  of  the  piston,  i.e.  the  plunger 
side  of  the  recoil  piston,  becomes  gradually  filled 

with  the  further  recoil.    The  pressure  in  this 
rear  chamber  however  is  zero  until  the  chamber  be- 

comes completely  filled.   If  Xs  is  the  displace* 
•ent  in  the  recoil  when  this  chamber  is  just  filled', 
obviously,  A(XS-S)=A'XS  hence 

Xs  =  A^T7"" 

Let  txs  =  the  corresponding  time  in  the  recoil. 
We  have  two  cases: 

(1)  IThen  Xs  <  E: 

(2)  Where  Xs  >  E. 

For  case  (1),  txs  and  Xs  are  connected  by  the 
equation, 

pob  (txs-to)* 

Xs»Xfo*[Vfo-  f-  ""-'o)-  7-77-7-;  KtM-t0)(ft> 

r  6"r(Vf~7fo) 

(approx) 

from  which  by  trial  values  we  may  estimate  txs 
For  case  (2)  we  aay  compute  txs  from 

2K(XS-E) 


T  *  "here  VX8  «   V? 


-xs  K 

~r 

Vr  »  Vf  approx.  =  max.  recoil  velocity  (ft. sec) 
K  *  total  resistance  to  recoil 
T  =  total  time  of  powder  period 
To  calculate  the  mean  pressure  in  the  chamber 
back  of  the  recoil  piston, - 

Dvl 
pj  =  where  D  =  density  of  the  fluid- 

53  Ibs/cu.ft. 

vm  =  the  mean  velocity  in  the 

Q  pipe 

and  v.  =  — ; r   ft/sec. 


713 


•here  Q  «  A(Lg-3)-A'Lb 

wa  »  area  of  pipe  (sq.ft; 

tb~txs  =  tine  of  travel  through  the  recoil 
displacement  Lb~Xg 

Now  tb  is  the  time  for  the  recoil  displacement 
Lb,  hence         mr(Vr-Vb) 


••here      /    2K(Lb-E) 
Vb  »  /?| 


"r 

To  calculate  the  maxifflum  pressure  in  the  chamber 
b.ck  of  tb.  pl.t..,-     D(,-A,)V. 

™"*   "'  " — lbs/s<-ft- 


where,  when  xg<E, 

Vxs  -  Vf  (approx)tbe  maximum  velocity 

of  free  recoil  (ft/sec) 
where  X,  >  E 

— _— __««-.^— _ 

/    2K(Xg-E) 
vxs  =  '  Vt     _ 


From  the  constrained  velocity  curve,  we  may  cal- 
culate p'  during  the  displacement  (Lg-Xs).  Since 
p >3po  approx.  we  may  assume  p1  constant  and  use 
the  previous  expression  for  pn.   It  is  important 
here  to  note  that  the  recoil  throttling  must  be 
modified  to  maintain  a  constant  pall  on  the  brake. 
Ph-pA-p'A1  (Ibs) 

D  Av   oi"* 

p-p'» (Ibs/sq.ft) 

2gC'.« 

Combining  these  two  expressions,  we  may  solve  for 
the  required  modified  recoil  throttling  area  wx 
(sq.ft)  in  terms  of  the  known  values  p'  and  Pb. 

It  is  important  that  the  recoil  brake  function 
at  least  at  the  end  of  the  powder.  We  place,  there- 
fore, 

S  »  B,  then  Xs  = 


713 


2K(Xa-E) 
=  ./  u*  _ 
xs 


mr 
and  tb  =  T+mr(Vr-Vb) 


2K(LS-E) 
Vb  =  A£  -        •    usually  p1  should  not 

mr       exceed  a.  few  hundred 

Ibs/sq.in.  and  the  value 

of  wa  and  A1  should  therefore  be  corresponding. 
In  such  a  case  no  material  effect  in  the  recoil 
throttling  is  obtained  and  a  modification  of  the 
grooves  is  unnecessary. 

DESIGN  OF  SIDE  PRAMS       The  loading  on  the 
GIRDERS.  girders  and  the  correspond- 

ing stresses  depends  upon 
the  method  proposed  for 
firing.   These  methods  may 
be  classified  as  follows: 

(1)     Firing  from  semi  fixed  base 

plate,  with  a  large  pintle  bearing 
and  the  girders  extending  to  the 
rear  supported  at  their  end  by  an 
outer  circular  track.  The  horizontal 
and  a  part  of  the  vertical  reaction 

is  transmitted  to  the  pintle  base 

a.  o  i  i  -- ;  ~J :  ;; 

plate,  the  horizontal  reaction  being 

taken  up  by  a  vertical  spade  extend- 
ing below  into  the  ground  from  the 
base  plate  and  the  vertical  load 
being  balanced  by  the  upward  re- 
action of  the  ground  on  the  base  plate. 
No  balancing  moment  is  assumed  to  be 

exerted  by  the  ground  on  the  base 
plate.   This  assumption,  makes  it 
possible  to  readily  determine  the 
upward  normal  reaction  of  the  outer 
circular  track.   We  have,  therefor*, 


714 


with  this  method  of  loading  the 
horizontal  and  vertical  reaction 
at  the  pintle  bearing  and  a  vertical 
reaction  at  the  tail  of  the  girder 
balancing  the  trunnion  reaction  due 
to  firing.   This  loading  should  be 
considered  at  both  horizontal  and 
maximum  elevation. 

(2)  Firing  from  the  pintle  base  plate 
assumed  bolted  down  to  a  concrete 
base.   In  this  method  no  outer  track 

for  supporting  the  tail  of  the  girder 
is  necessary.   We  have  therefore  at 
the  pintle  bearing  a  horizontal  and 
vertical  reaction,  together  with  a 
bending  couple  balancing  the  firing 
reactions  at  the  trunnions.   This 
loading  should  be  considered  at  both 
horizontal  and  maximum  elevation. 

(3)  Firing  from  a  special  layed  track, 
the  nount  recoiling  in  translation 

on  this  track.   By  this  method  the 
vertical  load  is  somewhat  distributed 
by  several  shoes  brought  down  in  con- 
tact with  the  track.  The  horizontal 
component  due  to  firing  at  the  trunnions 
is  balanced  by  the  total  sliding  friction 
equal  to  the  weight  of  the  mount  plus 
the  vertical  firing  component  times 
the  coefficient  of  track  friction 
and  the  inertia  resistance  of  the 
mass  below  the  trunnions  to  ac- 
celeration. Though  the  horizontal 
reaction  on  the  trunnions  is  theoretically 
slightly  reduced  due  to  the  acceleration 
of  the  cradle  in  which  the  gun  recoils, 
we  may  practically  consider  that  the 
total  firing  load  is  brought  on  to  the 


715 


trunnions,  since  the  acceleration 
of  the  total  mount  backwards  is 
relatively  small  and  the  mass  of 

the  cradle  quite  negligible  as  com- 
pared with  the  large  mass  of  the 
main  girders,  trucks,  etc.  below 
the  trunnions.   The  vertical  com- 
ponent due  to  firing  at  the 
trunnions  is  balanced  by  the  upward 
reactions  on  the  various  shoes. 
Finally  the  couple  produced  by  the 
horizontal  reaction  at  the  trunnion 
and  the  resultant  of  the  inertia 
resistance  and  the  shoe  frictions, 
is  balanced  by  a  couple  produced 
by  the  vertical  reaction  at  the 
trunnions  and  the  resultant  normal 
or  vertical  reactions  of  the  track 
or  guides  on  the  various  shoes. 

This  requires  a  uniforaiity  increas- 
ing upward  reaction  on  the  various 

shoes  towards  the  rear.   The  load- 
ings should  be  considered  at 

horizontal  and  maximum  elevation. 
(4)     Firing  directly  from  trucks 

riding  or  recoiling  back  on  the  rails, 
This  loading  is  similar  in  character- 
istics to  (3)  except  now  the  sup- 
porting reactions  are  concentrated 
at  the  truck  pintles.   Again  the 
loadings  should  be  considered  at 
horizontal  and  maximum  elevations. 
When  a  girder  is  designed  to  meet  all  four 
requirements  in  the  methods  of  firing,  we  have  for 
the  two  elevations,  eight  types  of  loading  to  be 
considered  as  applied  to  the  girder.   Knowing  then 
the  loads  brought  on  to  the  girde,  we  have,  the 
following  points  to  consider  in  the  layout  of  the 
girder  as  regards  its  strength. 


716 


(1)  The  proper  flange  area  to  carry 
the  requisite  bending  at  a  section 
of  given  depth. 

(2)  The  proper  depth  of  girder  for 

all  other  sections. 

(3)  The  proper  cross  section  of  the 
webs  for  carrying  the  total  shear. 

(4)  The  proper  pitching  of  the 
rivets  for  carrying  the  longitudinal 
shear . 

(5)  A  careful  study  of  web  reinforce- 
ments or  stiffeners. 

(6)  The  distribution  and  design  of 

cross  .beams  or  transoms  connecting 
the  two  girders. 

(7)  The  detailing  and  design  of  the 
pintle  bearing. 

(8)  The  reinforcement  in  the  web  re- 
quired for  the  elevating  pinion 
bearing. 

Reactions  between  tipping  parts  and  girder 

trunnion  reactions: 


2H=K  cos  0+E  cos  6^     (Ibs)  (1) 
2V-K  sin  0-E  sin  6^  +Wt   (lbs)(2) 
and  for  the  elevating  gear  reaction. 

Ks+Pbe 

E  =  ; In  battery  (Ibs)   (3) 

J 

Ks+»rb  cos  0 

E  *  Out  of  battery  (Ibs)  (4) 

j 
where 

H  and  V  *  the  horizontal  and  vertical  com- 
ponents of  the  trunnion  reaction 
(Ibs) 

K  =  total  resistance  to  recoil  (Ibs) 
E  =  elevating  gear  reaction  (Ibs) 
j  »  radius  from  trunnion  axis  to  line  of 

action  of  elevating  gear  reaction  -  with 
rack  and  pinion  =  radius  of  each  (in) 


717 


6£  =  angle  between  j  and  the  vertical. 
S  »  perpendicular  distance  from  line  through 
center  of  gravity  of  recoiling  parts 
and  parallel  to  bore  to  center  of  trunnions 
(in.) 

e  =  perpendicular  distance  from  axis  of  bore 
to  center  of  gravity  of  recoiling  parts. 
With  a  balancing  gear  introduced  between  the 
tipping  parts  and  girders,  we  must  modify  the 
trunnion  reaction  to  include  this  reaction.   The 
elevating  gear  reaction  is  not  changed,  since 
the  moment  of  the  tipping  parts  about  the  trunnions 
is  always  balanced  by  the  balancing  gear  in  the 
battery  position  of  the  recoiling  parts. 

Since  it  is  usually  customary  to  locate  toe 
trunnions  along  a  line  through  the  center  of  gravity 
of  the  recoiling  parts  parallel  to  the  bore,  S  =0, 
and  therefore,     p  e 

E  =  "-; —   in  battery 


/X        X>0   \ 
Wrb  cos  9 

g  m     .       out  of  battery 
j 

Now  since  e  is  usually  made  very  small, 

Pbe    Wrb  cos  0 

and         may  be  neglected  as  compared  with 

J        j     H  and  V.   Hence,  we  will  assume 

the  elevating  gear  reaction  to 

be  negligible,  and  we  have  the  total  firing  load 
brought  onto  the  girders  at  the  trunnion.  Then 

2H=  K  cos  0     :  approx.  reaction  between 
2N=K  sin0+Wt    :  tipping  parts  and  girder. 

Reactions  between  base  plate  and  girder. 

Considering  the  reactions  on  the  base  plate, 
if  it  is  considered  that  the  ground  can  offer  no 
bending  resistance  as  in  assumption  or  method  (1) 


718 


OF 


9.1-f 

.    e> 


Fig. 


719 


1 

A/ 

^-; 

//-*  —  ta  

\*-*7^Vv~ 

......                      1    \AS. 

1 

S^             VVy 

X 

^  VA> 

rfr,                      », 

3  ^^ 

i 

-c 
i 

-c 

J 

J7^^  .""Lfl- 

T 


i..l 


Fig.  3 


720 


of  loading  we  have  the  reaction  between  the  base 
plate  and  girder  as  equivalent  to:- 

(1)  A  vertical  reaction  through 
the  center  line  of  the  pintle 

bearing  *  V   (Ibs) 

(2)  A  horizontal  reaction  at  the 
pintle  Hp  (Ibs) 

(3)  A  couple  Hp(h-hp)(in.lbs; 
where  h  =  height  from  ground  to  trunnions  (in) 

hp  =  height  from  pintle  bearing  to  trunnion 

(in) 

In  method  (2)  of  loading  we  have  the  reaction 
between  the  base  plate  and  girder  equivalent  to:- 

(1)  A  vertical  reaction  through  the 
center  line  of  the  pintle  bearing? 

VP 

(2)  A  horizontal  reaction  at  the 

pintle  H 
P 

(3)  A  couple  resisting  the  over- 
turning moment  *  H  h 

P 

Constructively,  only  the  horizontal  reaction 
is  taken  up  at  the  pintle  bearing,  the  vertical 
or  normal  reactions  being  taken  up  at  the  travers- 
ing rollers.   Thus,  the  roller  reactions  are 
equivalent  to  a  couple  H  hp  and  a  resultant  vertical 
or  normal  reaction  V 

To  calculate  the  individual  traversing  roller 
reactions  we  proceed  as  follows: 

Consider  the  rollers  equally  spaced 
around  the  periphery  of  the  roller  path.   Then, 
taking  loooents  about  the  front  outer  or  end 
roller  in  the  direction  of  the  axis  of  the  bore, 
we  have,  for  the  various  roller  reactions,  see 
fig. (2). 

Assuming  "n"  chords  passing  through  a  pair 
of  rollers  and  perpendicular  to  the  axis  of  the 
bore  projected  in  a  horizontal  plane,  then, 


721 


pt=k(xt+y  )  pa»k(xa+y  )  ----  pn=k(xn+y) 
Taking  moments  about  the  front  roller 
k[2xt(xt+y  )+2x2(x2+y  )  ---  2xn-i+y=+xn(*n+y> 

=  H  hp+Vpr 
Simp  lif  ying,  we  have 

ky(2xt+2x2  --  2xn_1+xn)+k(2x*+2x2  ---  2x*_1+x*) 

-  H  hpvVpr 

and  for  the  summation  of  the  vertical  reactions, 
ky+Wk(xt+y)+2k(x?+y)  --  2k(xn_a+k(xn+y  )=Vp 

2k  ny+k(2xt+2x2+  ---  2xn_1+xn)=Vp 

To  solve,  we  note  that,  A(ky)+B(k)=H  hp+Vpr 

C(ky)+D(k)=V 
where  A  =(2xi+2xs  ---  2xn_x+xn) 

B  = 

C  =  2n 
D  =(2x 


Knowing  x^x  ---  xn  we  may  readily  obtain  Po»Pt  ---  pn 
To  compute  x^x  --  xn  for  the  rollers,  we  bave 

for  the  angle  to  the  various  chords, 

2n  360 

6  *  —  radians  or  -  degrees 
n 

_  2n  360  . 

9  -2  —  rad.  or  2  -  degrees 
a    n  n 

n  2*         n  360 
9n  =  2  ~  rad'  °r  2  ~ 

tbereforex»r(l-cos  6^)    (in) 

xa=r(l-   cos   92)    (in) 

xn=    r(l-   cos   9n)    (in) 

where  r  =  radius  to  the  center  line  of  the  roller 
path. 


722 


Proa  the  previous  equations  we  nay  now  compute  P0, 
Pt  -  Pn(lbs),  the  individual  roller  reactions. 

The  previous  formulae,  assume  contact  betiieen 
each  roller  and  the  roller  track  under  maximum 
firing  conditions.   If  the  roller  path  has  a  small 
diameter,  we  nay  have  the  condition,  when,  only 
the  rear  roller  is  brought  into  contact,  the  over- 
turning moment  on  the  girder  being  balanced  by 
a  couple  exerted  by  the  base  plate  an  upward  re- 
action at  the  rear  roller  contact  and  a  downward 
reaction  at  the  front  circular  clip  contact.   If 
the  circular  clip  has  a  radius  approx.  equal  to 
that  of  the  roller  path,  then  we  have  for  the 
sax.  roller  reaction  Hphp+Vpr=2pmaxr 

H  VV 
.av*   -  where  r  =  radius  of  the 


roller  path  (in) 
P>ax  =  max.  roller  reaction 
(Ibs) 

Vp=aax.  upward  reaction  at 
pintle  (Ibs) 


External  forces  exerted  on  the  girder  during 

firing: 

The  external  force  or  the  girders  are 
shown  in  plates  A  and  B  for  the  four  methods  of 
loading. 

In  method  (1)  of  loading,  we  have  the  re- 
actions of  the  tipping  parts  H  and  V,  the  reaction 
of  the  base  plate  H  and  V  together  with  the 
couple  Hp(h-h_)  and  the  reaction  of  the  outer 
track  on  the  tail  of  the  girder  N.  Further  we 
must  include  the  total  weight  of  the  girder  which 
though  actually  distributed  we  will  assume  con- 
centrated at  its  center  of  gravity  at  horizontal 
Ig  from  the  axis  of  the  trunnions. 


723 


Taking  moments  about  the  pintle  bearing, 

H  h  +H(h-h_)-Nl  =0  hence 

v      *  Hh 

N  =  ^   (Ibs) 

*n 

where  H  =  K  cos  £5  and  h  =  the  height  of  the 
trunnions  from  the  ground  (in). 

Knowing  N  we  may  compute  for  the  strength 
of  the  tail  of  the  girder,  for  method  (1)  of 
loading. 

In  nethod  (2)  of  loading  since  we  are  detail- 
ing the  strength  of  the  girder  in  the  region  of 
the  trunnion  and  pintle  reactions,  we  must  take 
the  actual  components  of  the  reaction  into  con- 
sideration.  These  consist  of  the  trunnion  and 
elevating  arc  reactions  of  the  tipping  parts,  that 
is  the  reactions  H,V,  and  E  and  the  reaction  of 
the  base  plate  consisting  of  the  various  roller 
reactions  and  the  horizontal  reaction  of  the  pintle 
as  shown  in  "Reaction  of  Base  Plate  on  Girder"  diagram, 

In  method  (3)  of  loading,  where  the  mount  slides 
back  on  a  special  constructed  track,  we  have  for 
the  reactions  on  the  girder. 

(1)  The  H  and  V  components  of  the 
trunnion  reaction  of  the  tipping 
parts. 

(2)  The  inertia  resistance  of  the 
girder,  resisting  tne  acceleration 
of  the  girder  acting  at  the 
center  of  gravity  of  the  girder  = 


dt« 

(3)  The  weight  of  the  girder  acting 

at  its  center  of  gravity  Wg 

(4)  The  normal  reactions  of  the  track 
shoes  Na  and  Nb 

(5)  The  frictional  or  tangential  com- 
ponents of  the  track  shows   n(Na+Nb) 


724 


In  calculating  the  stresses  on  the  various 
portions  of  the  girder  we  must  of  coarse  consider 

both  the  weight  and  inertia  as  distributed  forces, 
but  for  dealing  with  the  overall  reactions,  we 

may  assume  their   resultant  effect  as  concentrated 
force  passing  through  the  center  of  gravity  of 
the  girder. 

When  the  trucks  are  entirely  disengaged  in 
this  method  of  firing,  we  have, 

d2x 
H-nUa+M-"^  dT1  =  °    3nd  Na+Nb=v+V*g  when  tlie 

trucks  are 

not  disengaged  but  hang  from  the  girder,  we  must 

consider  both  their  weight  and  inertia  reaction, 
hence  if  Wtjj  =  weight  of  trucks  (Ibs) 

Mt-  =  mass  of  truck 
we  have, 


H-n(Ka-KTb)-(o)g+mtk)  -   and  Na+N 
dt* 

To  compute  Na  take  moments  about  Nb  (see  fig.  (4). 

d*x 
*.(Vlb)  +  H  h-Vlb-Vlb-Wg(lb-lb)-  mg  —  (h-hg)-0 

hence  H*X 

)+«  ~(h~  "  H  * 


N   =  -  —  --  -     (Ibs) 
a  la+lb 

and   for   Nb   talcing   moments    about   Na,    we    have 

-  . 


(h-hg) 


hence 

„     ,  -      -    (Ibs) 


In  method  (4)  of  loading,  we  have  the  mount 
recoiling  back  directly  on  the  rails,  and  the 
trucks  react  on  the  girder  with  reactions  Ha, 
N   and  Hjj,  Nb,  at  the  truck  pintles  a  and  b  .   The 


725 


tipping  parts  react  on  the  girder  with  components 
H  and  V  at  tne  trunnions.   In  addition  we  have  the 

inertia  resistance 

d8x 
nig  -  —  -   resisting  the  ac- 

celeration and  the 

weight  of  the  girder  both  acting  through  the  center 
of  gravity  of  the  girder. 

For  a  horizontal  motion  back  along  the  rails, 

tie   have 

d    x 
H  -(H.+Hv)-  m-  -  —  -  =  0     and    normal   to   the 

**   U  t       %*.*  I 

rails,  V+Wg-(VNb)  =  0 

To  calculate  Ha  and  H^  the  horizontal  components 
of  the  truck  reaction  we  must  consider  the  trucks 
separately.   In  firing  directly  from  the  rails 
the  trucks  are  usually  braked. 

If  W^jj  and  M^  =  weight  and  mass  of  either  truck 
Ww  and  f»w  =  weight  and  mass  of  a  pair  of  wheels 

I=«wka  =  moment  of  inertia  of  a  pair  of  wheels 

about  the  center  line  of  the  axle. 
d  =  diameter  of  a  car  wheel 

k  =  radius  of  gyration  of  a  pair  of  car 

wheels  (=0.7  d  approx.) 
NW  =  normal  reaction  at  base  of  car  wheel 

N«.  =  normal  reaction  of  brake  shoe  on  wheel 

»* 

per  pair  of  wheels 

fw  =  coefficient  of  rail  friction 
*  fs  =  coefficient  of  brake  shoe  friction 

Rw=tangential  force  exerted  by  rail  on  base 

of  car  wlieel 
Now  for  the  motion  along  the  rails,  we  have, 

dax 
Ha  -  S  Rw  =  mtk  j^ 

Considering  the  rotation  about  the  csnter  of  gravity 
of  a  single  wheel  we  have, 

2mwk2  d2x  4mwk2  d2x 


2Nsfs  wbere  n  =  no.  of 

pair  of  wheels   per 


726 


(3)  Of  LOAD/NG 


1 


te.«  v<— j 

I M  A//* 


MfT/iOD  (4)0f£  O4D/MG 

1     I/ 
^I^K 

•    ^^-£= 


A/- 


s-\  ^T 

•f  X-^HH — 


§ 


-9++ 


Fig.  4 


727 


SfCT/OA/  A-B 


o         o          o 

O  O  O 


T 


I 


e-j  ^*  I: 


Fig.  5 


728 


4rawk»  d*x 
truck  and  likewise  Hb=(mtk+n  —  —  )j^+SNsfs 


The  tera  2Ngfs  is  difficult  to  calculate  since  it 
depends  upon  bow  hard  the  brakes  are  set.   If  the 
brakes  are  set  to  skid  the  wheels,  no  rotation 
occurs,  and  we  have         (jax 

ZN   f 


Assuming    fB=0.2   and   2Nw=Hg+V+Wtk   we  have, 

dax 
Ha+Hb=2ntk   —   +  0.2    ("g+V+Wtk)    and   therefore 

d*x    H-0.2(Wt+Y+»tk) 

-  =  -  froffl  which  we  may  easily 

dt»      2mtk+mg       calculate  the  horizontal 

inertia  loading  for  any  position  of  the  girder. 

The  reactions  at  the  truck  pintles,  become 


res 

spectiveiy, 
b*    g^   b    1g'  + 

dax 
mg  (h-hg)-Hh 

/i  u  _  \ 

d2x 

'Btk  *     ~+Q-2   Na 
dt2 

Ubs; 
(Ibs) 

*b 

Hh+Vlj+Wgdg+1 

5                *      d  t  2      _        .»                     flVio^ 

\liOS  ) 

+0.2  Mb  (Ibs) 

Comparison  of  Truck  Pintle  Reactions. 
In  method  (3)  and  (4)  of  loading  we  find 


Nb-Na  =  -___——   Now  in  general  the 

horizontal  resistance 
is  small  as  compared  with  the  inertia  resistance 

nu  — r  Hence  we  may  approximately  assume  H=mg— 7 

8  dt            2H(h-hg)  Kdt 

therefore  Nb-Na  =  Further,  we  are  not 

la+^b      greatly  in  error  in 


729 


assuming  hg=  -   then  we  have,  Nb-Na  =  " i — 

1a*1b 

That  is  the  difference  of  load  thrown  on  the  rear 
and  front  truck   respectively  equals  the  horizontal 
trunnion  reaction  times  the  height  from  the 
trunnions  to  the  horizontal  center  line  through 
the  truck  pintles,  and  divided  by  the  distance 
between  the  trucks. 

Obviously  as  the  gun  elevates  H  decreases, 
while  V  increases;  therefore  at  max.  elevation 
the  loadings  on  the  trucks  are  more  nearly 
equalized. 

With  railway  carriages,  since  at  maximum 
elevation 

h 

H  is  relatively  small  compared 

with  Na  or  Nb,  for  all 

practical  purposes  we  may  consider  that  the  re- 
quired strength  of  the  girders  must  be  equally 
strong  on  either  side  of  the  trunnions. 


CHAPTER       XI. 
GUN  LIFT  CARRIAGE. 

Single  recoil  systems  where  the  recoiling 
mass  does  not  translate  in  recoil  parallel  to  the 
axis  of  the  bore,  appear  in  various  types  of  mounts. 
Illustrations  of  such  types  nay  be  found  in  our 
model  1897  Barbette  mount,  where  the  gun  and  top 
carriage  fora  a  single  recoiling  mass,  recoiling 
up  an  inclined  plane.   Railway  carriages  especially 
in  France  bare  been  used,  where  the  recoiling  mass, 
(gun  and  top  carriage  )  recoil  on  a  gravity  plane 
mounted  on  the  car.  The  object  of  the  inclined 
plane  is  to  return  the  piece  by  gravity  into 
battery.   Carriages  with  no  recoil  except  the  slid- 
ing back  of  the  gun  and  top  carriage  as  a  single 
mass  on  rails  have  also  been  extensively  used,  the 
resistance  to  recoil  being  merely  the  friction 
offered  by  the  rails  or  slides. 

CHARACTERISTICS  OP       Due  to  the  fact  that  the 
INCLINED  PLANE       recoil  is  not  along  the 
CARRIAGES.          axis  of  the  bore,  during  the 

powder  period,  a  component 
of  the  total  powder  force 
normal  to  the  inclined  plane 

or  slides  is  introduced.  This  component  therefore 
introduces  large  stresses  in  the  carriage,  the 
component  increasing  with  the  elevation.   The  ex- 
cessive stresses  thus  introduced  at  high  elevation, 
prohibits  the  use  of  this  type  of  mount  for  firing 
at  high  elevations  especially  for  large  calibers. 
The  type  of  mount  is  useful  for  where  the  elevation 
is  not  great.   With  large  size  howitzers  this  type 
of  mount  would  necessarily  produce  a  very  heavy 
mount  for  strength  and,  therefore,  from  the  point 
of  view  of  mobility  alone  could  be  regarded  as 
none  else  than  poor  design. 


732 


Since  the  gun  recoil  is  not  along  the  axis 
of  the  bore  a  reaction  on  the  projectile  normal 
to  the  bore  is  introduced.   This  reaction  reaches 
a  maximum  closely  at  the  maximum  elevation.   It 
possibly  introduces  unequal  wear  on  the  rifling 
in  the  gun  tube  itself.   This  reaction  further 
introduces  a  slight  spring  during  the  powder 
period  on  the  elevating  arc  and  pinion. 

APPROXIMATE  THEORY  OF   Even,  for  a  very  close 
RECOIL,  NEGLECTING    approximation  the  reaction 
NORMAL  REACTION  OF    of  the  projectile  normal 
PROJECTILE  ON  BORE.   to  the  bore  during  the 

powder  period  has  a  very 

snail  effect  on  the  recoil,  though  it  is  of 
importance  in  estimating  the  maximum  elevating 
arc  reaction  during  the  powder  period.   If, 
then  we  let 

Pfc  =  total  powder  reaction  on  base  of  projectile, 

in  Ibs. 
B  =  hydraulic  braking  of  recoil  cylinders 

parallel  to  inclined  plane  in  Ibs. 
R  =  total  friction  of  the  recoil  in  Ibs. 
wr  and  mr  =  weight  and  mass  of  recoiling 

parts  (in  Ibs) 
0  =  the  angle  of  elevation  of  the  axle  of  the 

bore 

6  *  the  inclination  of  the  inclined  plane. 
E  =  displacement  of  free  recoil  during 

powder  period  (in  ft.) 

T  *  total  time  of  powder  period  (in  sec.) 
Vf=  velocity  of  free  recoil  (in  ft/sec) 
K  =  the  total  resistance  to  recoil,  in  Ibs. 
b  =  length  of  recoil,  in  ft. 

Then  considering  the  recoiling  parts  during  the 
powder  period,  we  have, 

u  V 

Pbcos(0+6)-(B+R+Wrsin9)=  «r  r—  and  since 


dt 


K»B+R+WrsinO 


733 


734 


Pb   cos(2J+9)dt 
then  / 


" 


r 


but        Phcos(0+9)dt 

/—  -  =  Vf  cos  (0+9) 
n>r 

therefore  at  the  end  of  the  powder  period,  we  find 

KT 

Vr=Vfcos(0+9)-  —          (1) 
mr 

KT8 
and  Xr'J)  cos  (0+9)-  -  —     (2) 

2n  j, 

During  the  remainder  of  the  recoil,  we  have 

j  mrV*  =  K(b-Xr)  (3) 

Substituting  (1)  and  (2)  in  (3)  and  simplify- 
ing we  have 

£  mrVfcos2(0+9) 

K  =  -       (4) 
b-(E-VfT)cos(0+9) 

Obviously  Vfcos(0+6)  and  E  cos  (0+9)  are  the 
component  free  velocity  and  displacement  parallel 
to  the  inclined  plane. 

EXTERNAL  REACTIONS  ON  THE   If  we  consider  the  sys- 
RECOILING  PARTS  AND  TOP   tern,  of  the  gun  wg  and 
CARRIAGE  ROLLER  RE-       recoiling  top  carriage 
ACTIONS.  wc,  we  have  by  D1  Alemberts  ' 

principle,  considering 

inertia  as  an  equilibriating  force,  the  following 
external  reactions:- 

(1)  The  powder  reaction  along  the 
axis  of  the  bore  —  Pb 

(2)  The  inertia  force  of  the  re- 
coiling mass,  opposite  to  the 
motion  during  the  acceleration, 
and  in  the  direction  of  the 
motion  during  the  retardation 

and  parallel  to  the  inclined  plane  — 

d*x 

•r 


735 


(3)  Weight  of  the  total  recoiling 
parts  Wr 

(4)  The  normal  reaction  of  the 
rollers  E  N 

(5)  The  braking  pull  exerted  along 
the  axis  of  the  hydraulic  brake 
cylinder  B 

(6)  The  total  friction  along  the 
roller  track  R 

These  forces  are  shown  in  fig.(l) 

Pesolving  (1),  into  a  couple  and  a  single 
parallel  force  through  the  center  of  gravity  of 
the  recoiling  parts  and  combining  with  (2)  we 
have,(l)  and  (2)  equivalent  to, 

A  powder  pressure  couple  Pbd 

where  d  =  the  perpendicular  distance  between  the 

center  of  gravity  of  the  recoiling  parts 

and  the  axis  of  the  bore . 
A  component  parallel  to  the  inclined  plane 

through  the  center  of  gravity  of  the  recoiling 

parts dy 

Pbcos(0+9)mr —  =B+R+Y»rsinp=K 

and  a  component  normal  to  the  inclined  plane  through 

the  center  of  gravity  of  the  recoiling  parts  

Pbsii)(0+9)  Thus  (1)  and  (2)  reduce  to 

A  couple  Pbd   and  the  parallel  and  normal  com- 
ponents through  the  center  of  gravity  of  the  re- 
coiling parts,  K  and  Pbsin(0+9) 

To  reduce  the  couple  Pbd  and  the  consequent 
stresses,  the  center  of  gravity  of  the  recoiling 
parts  should  be  located  at  the  axis  of  the  bore, 
or  slightly  below  to  ensure  a  positive  jump. 
Since  the  center  of  gravity  of  the  gun  is  at  the 
axis  of  the  bore,  the  top  carriage  center  of 
gravity  should  also  be  located  at  the  axis  of 
the  bore.   This  is  impractical,  but  if  the  top 
carriage  is  made  light  as  compared  with  the  gun, 
its  effect  in  lowering  the  center  of  gravity  of 


736 


B 


I1  'I'  T  'i 


737 


the  total  recoiling  parts  is  small. 

To  compute  the  roller  reactions  on  the'  in- 
clined plane,  we  proceed  as  follows: 

Taking  moments  about  tlie  front  roller 
reaction  "0",  we  have  Khr+Pbd+Pblrsin(0+9)«fWr  (lr 

cos9-hrsin8)-Be=Nilj+N  1 

----  Nnln    (5) 

where  hr  and  lr  are  the  coordinates  normal  and 
along  the  inclined  plane  of  the  center  of  gravity 
of  the  recoiling  parts  with  respect  to  the  front 
roller  "0" 

e  =  the  moment  arm  of  B  with  respect  to  "0" 
Nnln  =  the  moment  of  the  n  th  roller  reaction 
about  "0" 

When  the  top  carriage  is  light  as  compared 
with  the  gun,  the  center  of  gravity  may  be  assumed 

approximately  at  the  trunnions  and  therefore  P^d^O 
Hence  (5)  reduces  to,  Kht+P^ltsin  (flf+6  )+Wr  (lt 

cos6-htsin  6  )-Be*  N^+N^  ----  Nnln   (6) 

where  htand  lt  are  the  coordinates  of  the  trunnion 
with  respect  to  the  front  roller  "0".  Further, 
we  have,  Pbsin  (0+0)  +Wrcos  9  =No+Ni+N2  ----  Nn  (7) 
If  we  assume  the  roller  base  is  rigid,  we  "have 
B«k(l+c)     N 


Therefore  if,  SMQ=  Htlt=   N2lg  -----  Nnln 

ZN  =  VV*.  --------  Nn 

we   will  have,   MQ=T<(1*+1*  +  1»    ------  1«)   kc(li  +  lf  +  l3  --  ln) 

(8) 
ZN*k(l1  +  la  ------  ln)  +  (n+l)kc 

From  which  we  determine  "k  and  c 

EXTERNAL  REACTIONS  ON  THE     If  we  consider  the 
MOUNT  AND  TRAVERSING  ROLLER   system  consisting  of 
REACTIONS.  the  gun,  and  top  car- 

riage, that  is  the 
recoiling  parts,  to- 

gether with  the  "bottom  carriage  which  rests  on  a 
circular  base  plate  supported  by  traversing  rollers, 


738 


we  nay  eliminate  the  Mutual  reaction  between  the 
recoiling  parts  and  bottom  carriage  since  it  has 
no  effect  on  the  equilibrium  of  the  system.   Further 
by  the  use  of  D'Alembert's  principle  we  may  again 
regard  the  inertia  resistance  of  the  recoiling 
parts  as  an  equilibriating  force. 
We  have  therefore  as  before, 

(1)  The  powder  pressure  couple  Pbd 

(2)  The  total  resistance  to  recoil 
through  the  center  of  gravity  of 
the  recoiling  parts  and  in  the  di- 
rection of  the  recoil K 

(3)  The  weight  of  the  system  Ws 

(4)  The  pintle  reaction  balancing 

the  horizontal  component  of  (2) 

(5)  The  traversing  roller  reactions. 
Let  Ws=  weight  of  system 

13  -  moment  arm  of  Ws  in  battery  about  rear 

traversing  roller 
lg  =  moment  arm  of  HS  at  recoil  X  or  b  from 

battery 

Wbc  =  weight  of  bottom  carriage 
lbc  =  moment  arm  of  tf^c  ?bout  rear  traversing 

roller 

Wr  =  weight  of  recoiling  parts 
l^.  =  moment  arm  of  Wr  in  battery  about  rear 

traversing  roller 
b  =  length  of  recoil 

The  moment  of  the  weight  of  the  system 
changes  during  the  recoil.   If  we  take  moments 
about  the  rear  traversing  roller,  we  have  for  the 
weight  during  the  recoil  WpdJ-Xcos  6  )*Wbclbc=«sls 
hence  Wslg=  Wsls  -  HrX  cos  6  and  when  X  =  the 
length  of  recoil  b,  we  have  Wslg  =  Wsls-Wrb  cos0 

Further  if,  h^  and  1^1  are  the  vertical  and  horizontal 
battery  coordinates  of  the  center  of  gravity  of  the 

recoiling  parts  with  origin  at  the  rear  traversing 
roller  then  the  out  of  battery  coordinates  become 

and  (l£-b  cos  9)  respectively.   We  have 


739 


for  the  nonents  about  0,  in  battery  W_l_-PKd-Khi 

_-_._---  8   S    D 

cos  6  -  K  lr  sin  6  +  Pbsin(0+e)(i£cos  6-  hr 
sin  6)+2Nilj+2Nslt  Nnln   (11) 

and  in  the  out  of  battery  posit.ion 

Wgls-Wrb  cos  e-K(br+b  sin  6)COs  6-K(kr-  b  cos  6)sin  6 

*  2  Ni1^2N2la Nnln   (12) 

If  we  assume  the  center  of  gravity  of  the  recoil- 
ing parts  at  the  trunnions,  then  Pbd  disappears, 
and  h£  =  h{  and  1^  »  1{   As  before  NQ«kc,  N  =k(lt+c) 
Nn=»k(ln+c)   hence 

9  II   1     j.  O  W   1     Ml     -l*fO12j.Ol2        1»\    / 1  *5  \ 

2Ni1t    +2Na1.  W«  *   k(2   1i*2   ll  '  ~1V      (13^ 

We   also  note    that  PbSin(0+6)cose-Ksin  6+l»8=  EN    (14) 

where  EN=k(2  lt+2  12  (21n_1+ln)+2kcn 

From  equation  (13)  and  (14)  we  may  solve  for  k, 
and  c  and  thus  eonpute  the  roller  reaction  NQ,^ 

INTERNAL  REACTIONS       With  gun  lift  mounts  the 
TRUNNION  REACTIONS,   trunnions  are  a  part  of  the 

gun  itself  and  are  located 
at  the  center  of  gravity  of 
the  gun.  Neglecting  the 

normal  reaction  of  the  projectile,  and  taking 
moments  about  the  center  of  gravity  of  the  gun, 
that  is  about  the  trunnions,  we  have,  E  j  =  0, 
(j-eonent  arm  of  E  about  the  trunnions),  there- 
fore the  elevating  arc  reaction  E  =  0.  If  Xt 
and  Yt  are  the  components  of  the  trunnion  re- 
action, parallel  and  normal  to  the  inclined 
plane,  respectively, 

w-  =  the  weight  of  the  gun  alone.  We  have, 
considering  the  gun  alone,  fig.( 


dV 
)-WgSin  9  -  mg  — - 


I 


(15) 
2Yt=Pbsin(0+6  )+Wgcos  6 

dV 

but   Pb   cos    (0+8)  -  K  =   mr  - 

dt 


740 


dV   "a  ag 

hence  ng  —  =  —  Pb  cos  (0  +  9)-  K  — 

1  dt   mr  mr 

Substituting  in  (15),  we  have 

°0      "ff 

2X     .  p.     cos    (0+9K1 ^)+K  -*•  -  Wesin  6  1 

flj  ID 

\    (16) 

9 


which  gives  us  the  components  of  the  trunnion 
reaction.   The  resultant  trunnion  reaction, 
"becomes, 

St  -  /  X£+Y$  (17) 

The  elevating  arc  reaction  is  zero,  except 
during  the  first  part  of  the  powder  pressure 

period. 

To  compute  this  "whipping  action"  during 
the  powder  pressure  period,  we  must  plot  the 
moment  of  the  normal  reaction  of  the  projectile 
about  the  trunnions  as  the  projectile  moves  along 
the  "bore. 

The  normal  reaction  of  the  projectile, 
equals, 

N  =m  —  sin  (01-6)  (18) 

dt 

The  weight  component  normal  to  the  bore  "being 

neglected  since  we  will  assume  a  fairly  large 
breech  preponderence,  but 

dv   P^cos  (0+6)-K   Pbcos(0+9) 
dt        mr  mr 

If  U  =  the  travel  up  the  bore 

Ut  *  the  distance  from  the  center  of  the 
projectile  in  its  initial  position 
to  the  center  of  the  trunnions. 

Then,  the  elevating  arc  reaction  becomes, 

N  (U-Ut) 
E  »  -* 

j 


741 

«Pb(U-Ut)sin2(0+9) 
=  -  -  -     (19) 
2<"rj 

From  a  plot,  the  maximum  moment  was  found  to  oc- 
cur, when  the  shot  reaches  the  muzzle,  and  we 

then  have  for  the  maximum  elevating  arc  re- 

action,  mPob(U0-Ut)sin2(0+e) 

E  =   °       -   (20) 


breech   when   shot 
leaves   muzzle) 


n    iic27  P|°ax    ,»   /,,   27  pmax. 

C  =    U(  ---  1)+  /  (1-  -~  -  )«-i   *    (twice 


16  pe  16  pe 


abscissa 
cf  max. 

pressure) 


Pe  =      (pjj    =total  max.  powder  force  on 

breech) 
VQ  =  muzzle  velocity;  Pe=  mean  powder  reaction 

on  breech 
Ifr  -  travel  up  bore  in  feet 

REACTIONS  ON  TOP     Neglecting  the  elevating  arc 
CARRIAGE.        reaction  during  the  powder  period, 
the  reactions  on  the  top  carriage 

reduce  to  the  following:— 

(1)  The  trunnion  reactions  divided 

into  Xt  an:*  ^t  an<*  equal  and 
opposite  to  the  component  re- 
actions exerted  on  the  gun. 

(2)  The  weight  of  the  top  carriage 
acting  through  its  center  of 
gravity  ---  Wc 

(3)  The  braking  pull  reaction  ---  B 
(4)     The  roller  reactions  of  the  in- 

clined plane. 

Assumng  the  center  of  gravity  of  the  top  carriage 
at  the  trunnions  for  convenience,  we  have 

dV 
2Xt-Wcsin  6  -  B  =  mc  — 


742 


2Yt+Wccos  6  =  £  N  (22) 

and  taking  moments  about  the  front  roller  reaction, 
we  have  2EMQ  =  2Xtht-2Ytlt+Wccos  6  lt-Wcsin  6.ht 

dV 
-  Be  -  mc  —  ht  (23) 

dv    [Pbcos(0+9)-K] 

where "   s  ~ 


"  at      ror 

ht  and  lt  are  the  coordinates  of  the  trunnion 
with  respect  to  the  front  roller 
"0",  and  normal  and  parallel  to 
the  inclined  plane. 

9  =  the  perpendicular  distance  from  the  front 

roller  to  the  line  of  action  of  B 
If,  as  is  usually  the  case,  the  center  of  gravity 
of  the  top  carriage  is  not  located  at  the  trunnions, 
we  have  equation  (21)  and  (22)  the  same,  but  equation 

(23)  modified  to:-                  Pbcos(0+8)K 
M0.2Xtht+2Ttlt+Wccoa  6  lc-Wcsin  9  hc  -[— ] 

»chc-Be          (24) 

where  lc  and  hc  are  the  coordinates  of  the  center 

of  gravity  of  the  top  carriage  parallel  and  normal 
to  the  inclined  plane  and  with  origin  at  the  front 

roller.   As  before,  the  moment  of  the  roller  re- 
actions 2Mo=Nt1i+N81a'l'Ns13 Nn1n 

therefore  2Mo=k(l»+l|+l* l*)  +  kc  (li  +  la+la ln) 

SN  =  klt+kla  kln+(n+l)kc 

and  Nn=kc,  N  »k(l  +  c),  N  =k(l  +c) N-^kdn+c) 

O     '    I       1     '    2       fc 

that  is  solving  for  k  and  c  we  determine  Nt  S^ 
Nn  knowing  the  total  normal. 

Substituting  in  Eq.(24), 
"g    mg 
mr    mr 

2Yt  aPbsin(0+e)*wgcos  e  and  noting  that,  niglt+mclc» 

mrlr 


743 


mght+IDchc  =  mrnr  we  have,Pb[  (ht-hr)cos(0+e) 


cos  6  -  Be  =  2  MQ  (25) 

Now  (ht-hr)cos(0+9)+(lt-lr)sin(0+9)  is  evidently 
equal  to  the  perpendicular  distance  between  the 
center  of  gravity  of  the  total  recoiling  parts 
and  the  axis  of  the  bore.   Hence  (25)  reduces  to 
Pbd+Pbsin(0+9)lr+K  hr-Wrhr  sin  8  +  Wrlr  cos  6  -  Be 

=  Z  M0  (26) 

where  d  =(ht-hr  )cos  (0+6  )+(lt~lr  )sin(0+9) 
This  is  evidently  the  same  as  equation  (5)  obtained 
in  the   consideration  of  external  force  on  the  re- 
coiling parts. 

REACTIONS  ON  BOTTOM  CARRIAGE.     The  reactions 

on  the  bottom 
carriage  consist 
of  the  following:- 

(1)  The  braking  pull  exerted  along 
the  axis  of  the  hydraulic  recoil 
cylinder. 

(2)  The  roller  reactions  normal  to 
the  inclined  plane. 

(3)  The  horizontal  reaction  exerted 
by  the  pintle  bearing. 

(4)  The  supporting  reactions  exerted 
by  the  traversing  rollers  in  a 
vertical  direction. 

Evidently  (1)  and  (2)  is  the  reaction  of  the  top 
carriage  on  the  bottom  carriage,  which  is  divided 
into  the  components  (1)  and  (2). 

Thus  in  battery,  the  moments  of  (1)  and  (2) 
about  "0"  the  point  of  contact  of  the  front  roller 
reaction  of  the  inclined  plane  reduce  to  ZMo+Be 
but  ZM0+Be=Pbd+Pbsin(0+9)lr+Khr-  Wrhr  sin  8+»rlrcos  9 

where  ir  and  hr  are  the  coordinates  of  the  center 
of  gravity  along  and  normal  to  the  plane  of  the  re- 


744 


coiling  parts  with  respect  to  the  front  roller. 

Therefore  during  the  powder  pressure  period 
the  reaction  of  the  top  carriage  on  the  bottom 
carriage  is  equivalent  to, 

(1)  A  powder  pressure  couple  "Pbd" 

(2)  A  component  of  the  powder  force 
normal  to  the  inclined  plane  and 
through  the  center  of  gravity  of 
the  recoiling  parts  nPb  sin  (0+0)" 

(3)  The  total  resistance  to  recoil 
parallel  to  the  inclined  plane, 
and  through  the  center  of  gravity 
of  the  recoiling  parts  "K" 

(4)  The  total  weight  of  the  recoil- 
ing parts  through  the  center  of 

gravity  of  the  recoiling  parts 

ii  ui  " 
wr 

During  the  pure  recoil  or  subsequent  retardation, 
we  have,  2M0+Be=Khr-Wrhrsin  9+Wrlr  cos  6 
and  therefore  the  reaction  of  the  top  carriage  on 
the  bottom  carriage,  is  equivalent  to 

(1)  The  total  resistance  to  recoil 
parallel  to  the  inclined  plane 
and  through  the  center  of  gravity 
of  the  recoiling  parts  K. 

(2)  The  total  weight  of  the  recoil- 
ing parts. 

To  compute  the  horizontal  pintle  reaction,  we 
have  H  »  K  cos  0  -Pb  sin(0+9)sin  0  the  total 
normal  reaction  on  the  traversing  rollers,  be- 
come ZN*Ptsin(0+9)cos  9  -  K  sin  9  +1"r'f*bc 
where  W^c  =  weight  of  bottom  carriage 
If  further  1£  *  moment  arm  of  »r  in  battery  about 

rear  traversing  roller 
x  «  recoil  displacement  from  battery 
lbc=  "oment  arm  of  Wbc  about  rear  traversing 

roller 

Ws  =  weight  of  entire  system  above  traversing 
rollers 


745 


lg=  moment  arm  of  Wg  about  rear  traversing 

roller 

Then,  for  the  moment  of  the  weights  about  the  rear 
traversing  roller,  we  have, 

Wr(lp-  x  cos  0)+*bc1bc  =  lf's1s  ~  wr  x  cos  ^ 
Therefore,  for  the  moments  about  the  rear  travers- 
ing roller,  we  have 
ZM0=Wsls-Wrx  cos  0-Pbd+Pbsin(0+6)[ (l£-x  cos6)cose 

-(h£+  x  sin8)sin  Q]  -K( (h£+  x  sin8 )cos9-(l£-x  cos6)sin8] 

When  Pb  is  a  maximum  x  is  negligible;  therefore  for 
the  maximum  roller  reaction,  we  have 

ZM0=  Wsls-Pbd+Pbsin(0+9)[l^cos8-h;sine]-K[h^cos6- 
EXACT  THEORY  OF  RECOIL  Doe  to  the  normal  reaction 
CONSIDERING  NORMAL      of  the  powder  charge  and 
REACTION  TO  BORE  OF     projectile  during  the  travel 
PROJECTILE.  up  the  bore,  the  recoil  is 

more  or  less  effected,  de- 
pending of  course  on  the  weight  of  the  shell  and 
powder  charge  as  compared  with  the  weight  of  the 
recoiling  parts.   Let 

Pb  =  powder  reaction  on  breech  of  gun 
P_=  powder  reaction  on  base  of  projectile 
Pe  -  mean  powder  reaction  in  bore  of  bin 
N  ^normal  reaction  of  projectile  to  axis  of 

bore 
N"t=  normal  reaction  of  powder  charge  to  axis 

of  bore 

N  =  N  +  N  =  the  total  normal  reaction  of 
powder  charge  and  projectile 
to  axis  of  bore. 
B  +  R  =  total  braking  resisting  recoil 

parallel  to  inclined  plane, 
w  and  B  =  weight  and  mass  of  projectile 
wr  and  mr  =  weight  of  mass  of  recoiling  mass 
w  and  m  =  weight  and  mass  of  powder  charge 
0  =  the  angle  of  elevation  of  the  axis  of  the 
bore 


746 


9  *  the  angle  of  inclination  of  the  inclined 

plane 
x1  and  y*  =  coordinates  along  and  normal  to 

tbe  axis  of  the  bore. 
a  =  travel  of  the  projectile  along  tbe  bore 

or  relative  displacement  along  tbe  axis 

of  tbe  bore 
x  =  tbe  projection  or  component  of  the  absolute 

displacement  of  the  projectile  parallel 

to  the  inclined  plane 
Considering  the  motion  of  the  projectile,  we  have 

d*x  ' 
Pp  -  n  —  +  ng  sin  0 

N  =  •  -  sin  (0+6)+«g  cos  16  (2) 

dt» 

where 

d«x'  d«u   d«x     ,.,  -.          ,_. 

T  -  T  -  ?  ««•  <«*•)       «) 


for  the  motion  of  the  powder  charge, 


"  PP  =  [    ~  2     cos(0+e>1+5«  sine) 


N 
* 


ird*x'   d*x 
=  ytr—;  --  r-T  CO8  (0*8)k»g  sin  «J   (4) 

*  dt*    dt 

5  ^  sin(0+e)*Ig  cos  A  (5) 

4  1 

where 

.  2 


(6) 


Is  the  resultant  acceleration  of  the  center  of 

gravity  of  the  ponder  charge,  and  for  tbe  action 

of  tbe  recoiling  parts, 

Pbcos(0+e)-N  sin(0+e)-«rg  sin  6  -(B+R)-«P  ^ 

Nbere  N  «  Ht+N, 

Combining  the  above  equations,  we  have 


747 

I.d«x'          I  d*x 

(«)•*•-;  ...  cos(0+o;-  -  T—rcos2  (0+6  )  +  (m+m)g  sin0cos 
e,   at*          2  dt* 

d*x 

(0+9)-(m+I) sin*  (0+9)-(m  +  [B)g  cos0sin  (0+6  )-ni_g  sin9 

dt« 

d«x 
-(B+R)=»r  — 

Expanding  and  simplifying,  we  obtain 
•  d'x1         5  dgx  _   I  d'x  . 

dax 
-(m+I+iBr)g  sin  9  -  (B+R)  *  mr  — - - 

U  t- 

that  is 

n  .  ,d*x  *         d*x  .  m.  dflx 


=B+R*(B+I+«r)g  sin  9  (8) 

It  is  to  be  noted  that 

d*x'         d*x 

[    cos  (0+9)-  -—rsin*  (0+9)]  is  the  projection  or 
dt  *          dt 

component  of  the  re- 

suotant  acceleration  of  the  projectile  parallel 
to  the  inclined  plane,  and  B+R+(m+mr+in)g  sin  9 
is  the  total  external  force  parallel  to  the  plane. 
Neglecting  gravity  and  with  free  recoil(B+R=0)f 
that  is  no  extraneous  force  acts,  hence  we  have 


Ma 

In   terms   of   the   relative  acceleration  *—  * 

d  t 

d»x'  d»u   d»x       . 
since 


i.d"u  .    d«XT    .     m.d*x 

[  (.*-  Jcos  (0+8  )  +—  ]  »  (•+) 


hence 

(*n+")T-T  cos(0+9)=(m+B+»r)7f7  (10) 

2  dt2  dt1 

and  by  integration 

<•*?    )J7T  cos    (0+9)=(m+iii+inr)^  (11) 

cat  a  t 


748 


(a+-)u  cos  (0+9)=(a)+m+mr)x  (12) 

Hence  (10,  (11),  and  (12)  gives  us  the  free  ac- 
celeration, velocity  and  displacement  up  the  inclined 
plane  with  respect  to  the  corresponding  function 
up  the  bore  of  the  gun. 

Again  considering  equation  (8)  and  substituting 

d«x'   d«u   d«x 

—  —  =  -  -  —  cos  10+9) 
dt«    dt«   dt« 

we  have 

m  d*u  d*x 

(m+«)T~7  cos(0+9)-(B+R+lm+fii+mr]g  sinQ  )-(m+m-»-in..)  —  r  =  0 
2  at*  *  d  t 

hence 


dtx  dtu  B+R(m+S+ni  )g  sin9 

(13) 
dt*    n+I+mp 

Integrating, 
+ 


du          B+R+(m+m+mr  )g  sin  6 
—  =(  -  )--  coa(0+9)-[  -  =  -  ]t   (14) 
dt 


and 

m 

B+R+(m+i+mr)gsin   9 

x  =  (—  =  -  )u  cos(0f9)-  -[  -  =  -  ]  t*  (15) 
B-HB  +  nij.  m+m+mr 

which  are  the  general  equations  of  constrained  re- 
coil during  the  travel  of  the  projectile  up  the 
bore. 

Neglecting  m  and  ro  as  small  compared  with  mr 
and  if  we  let  m 


then 

dx  B+R+mPg  sin  9 

—  »  Vf  co8(0+ot)-(  -  -  -  )t 

dt  a 


B+R+Bpgsin6 

x  =  E  cos  (0+«)  -  i  (  -  1  -  )  t*        (15') 

"r 

which  are  sufficiently  approximate  for  ordinary 


749 


calculations . 

HUMERICAt  COMPUTATION. 

10"  Gun  Barbette 


K  -  braking  force 


cos  (0+6)* 

(0+8)  *  190 


2  b+T  Vfcos(0+6)-E  cos(0+6) 

89.820(29.76x0.9455)' 

2  (—+0.0446x29. 76x0. 9455-0. 9455)32. 2 

L  & 


247000  Ibs. 


89 . 820 (29 . 76x0.9976)' 
5Q  — •  274000  Ibs, 

32.2x2(—  +.0446x29.76x0.9976-0.9976) 

1  o 

Wr  »  89820  Ibs. 
Vf  *  29.76  ft/sec, 
b  *  50  in. 

T  »  0.0446  sec. 
E  »  1  ft. 

Zero  elevation  0  »  0° 
9  »  4° 


TRUNNION  RIACTIONS. 


0*19° 


750 


Y   -    F   sin(0+6)+Wg    CoS    6 

fl         K 

x  «  F  cos  (0+e(i-  -SO+K  -*.r  fi  sin  e 

Mr    mr 

K  »(Wr  sin  6+B)  -  247000  Ibs. 
K  =  PmxA  -32000*78.54  =  2513000  Ibs. 
Wg  =  76830  Ibs. 

As  a  check,  we  may  consider  the  forces  external 
to  the  system  above  the  rollers. 


F  cos(0+9)ht+F  sin(0+e)Lt+Wrcos  8Lt-Wrsin  ht-[P  cos 
(0+e)-K]ht-Be  *  ZM0 

2376000  "31    «  73,656,000 

2513000  "  .3256  *  Ib  *      12,273,500 
89820  x  .9976  x  15  =         1,344,000 


87,273,500 

89820  x  .0698  x  31  *  194,000 

2376000-247000  x  31  »       66,000,000 
240700  x  12  «  2,889,000 

69,083,000 

ZM0  «  18,190,500  moment  of  the  rollers 
ZN=F  sin(0+6)+Wrcos  6  =  2513000*  .  3256+89820*  .9976 

»  907600  Ibs.  total  normal 
load  on  the 

7fift^0 

X  *  2376000(1-   2)-76830x.  0698^247000 


343570-5.363  +  211284  =  549500  Ibs. 


751 


Y   =   818000+77000   »    895000   Ibs. 


5Z5.000 


in 

C\J 


GUN 


Sectional   Modulus=195 


Force  on  the  trunnions  * 


*  10s   X30.2+80.1 

*  10«  /1.  103 

=  1.050000  Ibs. 


525000*.  3.  375 

S  =  -  =  9080  Ibs/sq.in.  fibre  stress 
195 


ROLLER  RBACTION. 


about  front  roller, 


F  cos0-K 


752 


B  »  K-Wrsin9  »  B  »  240700 

2M(o)»  Xht+YLt+Wccos  9  Lt-Wcsin  9  nt-«c  ^f  bt-  Be 


d'x 
dt« 


549500x31+895000xl5+13000x.9976xl5-13000x 

.0698*31  -  (2376000  -  247000)  130°°  x  31  - 

89820 
240700x12  *  18185000  moment  on 


V+Wccos  9 


the  rollers. 

895000+13000X.9976  -  907960  Ibs. 

total  normal 
load  on  rollers 

jigACTIOM     OK     TH8     TRAVgRSING     BOLLIR8. 

F  sin(0-e)+WtWs-Ksin  9 

(P  cos  0-K)cos  6  +F  sin  (£l-«-6) 

wt.  of  reooiling  part  89000 

Mt.  of  the  rest  53000 

2513000X. 2588+89000+53000 (-247000). 0698 

650000+89000+53000-17200  =  774,800 

(2513000x9455-247000). 9976+2513000X. 2588 

2124000  +  650000   *  2774000 


667,600 
53,000 


AO"O^V ,-   I 

r^r  ~r~r  T  L 

VV   )jh L   i4   ln 


20  rollers 


117,000" 


753 


SECTION  MODULUS 
537, 


50,000 


M(0)-722000x80+53000x65  »  61450000  inch/lbs. 

moment  on  the  rollers. 

M  «K(2  1J+2  If  lg)+Kc(2  lt+21t 2  ln-x+1,,) 

V  «  XN-K(2  lt+2  lf  2  ln_1+ln)+2Kc 

6145000»K(2x2"73.+2x974  +2x2072  +2x34.4  +2x49,. 5 

+2x64.9  +2x78.8  +2x89.6  +2x96.4  +99  )+Kc 
(2x2.3+2x9.4+2x20.2+2x34.4+2x49.3+2x78.8 

+2x89.6+2x96.4+99) 
hence  6145000=73600K+990Kc 

774800"   990K+  20Kc 


61430000 
34000000 


73600K+990KC 
49000K+990Ko 


27450000  »  24600K 


Hence  8  -  1180 


Reaction  on  the  last  roller 

99x1180=117000  Ibs. 

Force  due  to  rifling  and  its  effect  on  the  travers- 

ing chain. 


Frt  =  Iw 


rt 


MK*n 

rt 


2  M 


754 


v  »  5  in. 
t  -  .0162 
«  .  60S 

32.2 

R  =  .8r  =  4  rifling  1  turn  in  c5  caliber 
1  turn  in  250  inches  *  20.83  ft. 


"  *  2  " 


606x4*x770x2 


77°  ""•• 

606x16x770 


32.  2x5x.  0162x12 
238500  Ibs. 


32.  2x5x.  0162*12 


7465920 

- 

312984 


Torque  »  238500x4  »  95400  ft.  Ibs, 


95400x  sin  15°  *  9540Qx.2588  =  24700  ft.  Ibs. 
"  TR  sin  0  =  24700 


TD  «  —  x  24700  «  -  2500  Ibs. 
49 

Tension  on  the  chain  at  pinion 


2500 


500  Ibs. 


* 

. 


755 


VELOCITY  OF  FHCC  HKOL 


mnfi  offnojfcn.f-X3.7i 

TfVnfLfD  BY  RECOLIH(,f*KTf  DURING 

z.ssr  re. 


KccotLiNs  mxrs 


OfPHOJECr/tt 


/O-INCH  BARBETTE  CAfff?/AG£ 

MODEL  OF  /893 
THE  EFFECT  OF.  THE  TRAVEL  OF 


THE  PROJECT/LE  UP  THE 
BGHE  ON  THE  ELEVAT/MG  ARC 


CHAPTER   XII. 
DOUBLE  RECOIL  SYSTEM. 

OBJECT    In  order  to  reduce  the  reaction  of  recoil 
on  a  carriage  to  a  moderate  value  when  the 
caliber  is  large  a  long  recoil  is  necessary. 
A  long  recoil  requires  long  guides  and  in 
addition  is  usually  prohibitive  due  to 
breech  clearance  necessary  to  avoid  a  great  loss 
in  stability  due  to  the  overhanging  of  the  recoil- 
ing weights  at  low  elevation  when  the  gun  is  out 
of  battery,  etc.   A  long  recoil  may  be  avoided  by 
the  use  of  a  double  recoil  system  and  the  stability 
of  a  railway  or  a  caterpillar  carriage  at  the  same 
time  increased.   This  latter  factor  is  the  real 
distinctive  value  of  a  double  recoil  system  over 
a  corresponding  single  recoil  system. 

It  is  important  to  note  that  a  caterpillar 
or  railway  car  braked  with  a  single  gun  recoil 
system  is  essentially  a  double  recoil  system,  the 
ground  or  rail  offering  a  tangential  reaction  which 
corresponds  to  the  reaction  of  the  lower  recoil 
system. 

Obviously  when  a  top  carriage  moves  up  an 
inclined  plane  under  the  recoil  reaction  of  the 
gun  and  the  resistance  of  the  lower  recoil  system 
or  when  with  a  single  recoil  system  the  cater- 
pillar or  railway  car  runs  back  on  the  ground  or 
rail  under  the  recoil  reaction  of  the  gun  and 
the  resistance  of  the  ground  or  rail,  the  recoil 
reaction  of  the  gun  becomes  different  and  the 
throttling  grooves  must  therefore  he  necessarily 
different,  then  with  a  single  recoil  system  when 
a  constant  recoil  reaction  is  imposed  between  the 
gun  and  top  carriage. 

757 


758 


CLASSIFICATION.    In  the  design  of  a  double  recoil 
system  it  is  desirable  in  order  to 
simplify  calculation  and  secure 
uniformity  of  stresses  throughout 
the  recoil  to  have  both  the  upper 
and  lower  recoil  reactions  constant  throughout 
recoil.    However,  in  ordnance  design  it  has  been 
customary  to  mount  single  recoil  mounts,  gun  and 
top  carriage  together  on  caterpillars,  etc.,  and 
for  augmenting  the  stability  to  allow  the  top 
carriage  to  recoil  as  well  up  an  incline  plane, 
the  inclination  of  the  plane  being  sufficient  to 
bring  the  systeu  into  battery  after  the  recoil. 
The  recoil  reaction  of  the  upper  system  can  there- 
fore, with  a  double  recoil  no  longer  be  constant 
since  the  recoil  reaction  is  the  sum  of  the  air 
reaction,  a  function  of  the  relative  displacement 
between  the  gun  and  top  carriage,  and  the  throttling 
reaction  which  is  a  function  of  the  relative  velocity. 
Therefore,  with  a  constant  braking  on  the  lower  re- 
coil system,  to  ascertain  the  displacement  of  the 
top  carriage  up  the  incline  plane,  it  would  be 
necessary  to  carry  on  a  somewhat  elaborate  point 
by  point  integration  for  the  various  dynamical 
equations  and  displacements  at  each  point  of  the 
recoil. 

Hence  in  the  following  discussion  we  will  con- 

• 

sider  the  dynamical  relational- 
CD     With  a  constant  resistance  for 

both  upper  and  lower  recoil  systems. 
(2)     With  a  given  upper  recoil  system 
and  a  constant  resistance  for  the 
lower  recoil  system. 

APPROXIMATE  THEORY  FOR  (1).     Reactions  and 

velocity  for  double 
recoil  systems:   Let 

P  =  resistance  of  gun  recoil  system 
W or  wr  *  wt.  of  recoiling  parts  (upper) 


759 


O/V 


SYSTEM 


Fig.l 


760 


WgOr  wc  =  wt.  of  top  carriage  and  cradle  (lower) 

V  =  initial  velocity 

Z  *  displacement  of  gun  on  carriage,  i.  e.  = 
relative  displacement 

N  =  upper  normal  reaction  between  recoiling 
parts  and  top  carriage 

M  -  lower  normal  reaction  between  top  car- 
riage and  inclined  plane. 

X  =  total  run  up  on  inclined  plane. 
/  or  v  =  velocity  of  combined  recoil 

t  =  corresponding  time  for  combined  recoil 

0  =  angle  of  elevation  of  gun 

6  =  inclination  of  inclined  plane. 

Since  during  tbe  powder  pressure  period,  there 
is  no  appreciable  movement  of  the  top  carriage  up 
the  inclined  plane,  and  the  timeaction  of  both 
the  upper  and  lower  recoil  reactions  is  negligible 
as  compared  nith  their  time  actions  in  tbe  pure 
recoil  period  after  the  ponder  period,  we  may  as- 
sume the  recoiling  mass  to  have  an  initial  velocity 
V  at  the  beginning  of  the  recoil,  where 

wv0+«  4700 

V  =  0.9  ( ) 

"r 

where  w  =  weight  of  projectile 
w  =  weight  of  charge 
v0  =  muzzle  velocity 

and  0.9  is  a  constant  to  allow  for  the  effect  of 
the  recoil  reaction  on  the  recoiling  mass  during 
the  powder  pressure  period.   Consider  now  fig.(l) 
Tbe  retardation  of  the  recoiling  parts  is  the 
vector  difference  of  tbe  velocities  at  the  end 
and  beginning  of  tine  t  divided  by  "t",  that  is 

v-V 
a  -        hence  assuming  axes  parallel  and  normal 

to  the  guides  of  the  upper  recoiling 
parts,  we  have  the  following  equations  of  motion 
for  the  recoiling  parts, 


761 


g 

snd        «       ,_  ^x 
^   wr  v  sin(9+0) 
N-Wrcos0  = -5 '—  (2) 

g     t 

Since  tliere  is  no  roation  the  cou  pie  between  the 
recoiling  parts  and  top  carriage  need  not  to  be 
considered. 

Next  considering  the  motions  of  the  carriage 
above,  we  have,  along  the  inclined  plane: 

H, 

„  i  „   O   D   _   . 

C1 


P  cos(0+9)-N  sin(0+6)-Wcsin  6  -  R  =  —  -     (3) 


and  normal  to  the  inclined  plane, 
N  cos(0+6)+Wccos  9  +p  sin(0+9)-M  =  0        (4) 
If,  after  the  recoiling  mass  and  top  carriage 
are  brought  to  a  common  velocity,  we  consider 
both  as  a  single  mass  in  motion  neglecting  the 
effect  of  the  further  motion  of  the  gun  on  its 
slide,  the  common  mass 

brought  to  rest  by  a  constant  force  H. 

g 

Hence  the  retardation  after  time  t,  becomes, 

Rg 
ar  =  -j — *r   and  the  interval  of  common  retardation, 

r  c  becomes,     ^  +^ 

tr  =  and  the  corres- 

yj  +w         "g    ponding  displace- 
t.        r  c 
ment  -  artr=  v*  •   Therefore,  the  total  dis- 

"g       placement  (since  the  top 
carriage  is  uniformly  accelerated  to  a  velocity 
v  at  time  t)becomes. 


Since  the  relative  displace- 
ment equals  the  absolute 
displacement  of  the  gun  parallel  to  the  guides 
minus  the  displacement  of  the  top  carriages 
parallel  to  the  guides,  we  have 

V+v  cos (0*9)     v  cos  (0+9)  V 

Z  = t t  hence  Z  *  -  t 

22  2 


762 


763 


ENSRGY  SQUATION  FOR     Let  x1  and  y1  =  the  co- 
DOUBLE  RECOIL.        ordinates  parallel  and  nornial 

to  the  gun  axis. 

x  and  y  =  the  coordinates  parallel  and  normal  to 
the  top  carriage  inclined  plane. 

v 

xt  =  -j-  t   where  xx  =  the  displacement  of  the  top 

carriage  up  the  inclined  plane 
at  the  instant  when  the  re- 
coiling mass  and  top  carriage 
icove  at  common  velocity  v, 

hence  xt  =  -  t  .   Then  for  tne  recoiling  parts,  we 

have,  (P-Wr  sin0)x'=  i  [ V*-v2cos2 (0+6 )]   (!')  (In 

direction  of  upper  guides)  and 

(N-Wrcos)xtsin(0+9)  =  ^rorv2sin2(0+6)     (21)  (at 

right  angles  to  upper  guides),  and  for  the  top 
carriage  alone,  we  have 

[P  cos(0+9)-N  sin(0+6)-Wcsin  6-R]xt  =  \  Mcv2   (3 ' ) 
(Top  carriage  up  plane) 

Subtracting  (31)  fros  (I1),  we  have 
P[xl-xtcos(gJ+e)]-wpsinefx'  +  N  sin(0+6)xt+Wcsin  8.xt 

+Rx«  in.fV'-v^os8  (0+6)]-  -  0  v2 

S   f  2     t» 

and  substituting    (2)   in    (4),    we   have 
P[x'-xtcos(0+9)-Wrsin0.x'+  ^mrvasin2 (0+6)    +Wrxtcos 
0sin(0+Q)-  ~  mrtV*-vacos2 (0+6)]+  i  mcv2    +   wc   sin 

9.x+R  xt  =  0 
Now  the  relative  displacement  between  the  gun  and 

top  carriage  becomes,  Z  =  x'-xt  cos  (0+6).  Hence 

the  above  expression  reduces  to 

PZ-Wr[x'  sin0-xtcos0sin(0+6)]+  |mrva+  %cv*+Wc  sin 

e.x+R  xt  =  |  mrva  (7) 

Now  Wr[x 'sin0-xtcos0sin (0+6)]  is  evidently  fhe 
work  done  by  gravity  on  the  recoiling  parts  and 
Wc  sin  9.x  is  the  same  for  the  top  carriage.   In 
terms  of  the  relative  displacement  Z,  the  work  done 
by  gravity  on  the  recoiling  parts  may  be  obtained 
by  consideration  of  fig. (2) 

From  fig.(  2),   we  have,  x '»Z+xtcos (0+6) 


764 


and  the  work  done  by  gravity  on  the  recoiling 

parts  becomes,  Wp(Zsin0-xtsin  6) 

=  Wrtx'-xtcos  (0+9)sin0-xt  sin  6] 

*Wr(x  'sin0-xtsin0cos  0cos  9+xtsin0sin  9-x  sin  6) 

*  Wrt*  'sin0-x^sin0cos0cos  6+x  sin  6  (sin*0-l] 

=  Wr  (x  'sin0-xtsin0cos  0-xtsin  6  cos2  0) 

=  tfr(x'sin0-xxsin(0+9)cosen 

Hence  equation  (7)  reduces  to, 

PZ-Wr(Zsin0-xtsin  9)+Wrxtsin  9+i(mr-nn 


v 
where  xt  =  -  t  .  Further  since  R(X-XI)=  7(<nr+mc)v2, 

equation  (8)  reduces  to 

PZ-Wr(Zsin0-xsin9)+Wc  x  sin  6  +RX  =  J  mrva    (9) 
Equation  (8)  is  almost  obvious  from  the  theory  of 
energy,  since  the  total  initial  energy  -  mrv*  plus 

the  work  done  by  gravity  Wr(Zsin£J-xtsin  9)-Hcxtsin  9 
equals  the  final  Kinetic  energy  of  the  system 
-(mr+mc)v*  plus  the  work  done  in  the  upper  and 
lower  recoil  brakes  PZ  +  Rxt  to  the  combined  re- 
coil. 

Equation  (9)  is  also  self  evident  since  the 
final  Kinetic  energy  of  the  system  equals  zero  after 
the  system  has  recoiled  the  total  displacement  x 
up  the  inclined  plane. 

RECAPITULATION  OF  APPROXIMATE     When  the  re- 
FOJRMULAE  fOR  DOUBLE  RECOIL     sistance  to  recoil 
BRAKES.  is  assumed  constant 

for  both  upper  and 
lower  recoil  systems 

we  nay  with  a  very  close  approximation  obtain  the 
principle  reaction,  by  the  previous  derived  formulae. 

These  formulae  are  recapitulated  in  the  following 
group  for  convenience  in  calculation.   Then  if, 
w  =  weight  of  projectile 
w  =  weight  of  charge 
v0  »  muzzle  velocity 


765 

Wr  or 

wr  *  weight  or  recoiling  parts 
w0  =  weight  of  top  carriage 
Z  =  displacement  of  gun  on  carriage 
X  =  total  run  up  on  inclined  plane 
If  =  upper  normal  reaction  between  parts. 
0  =  angle  of  elevation  of  gun 
9  =  inclination  of  inclined  plane 
V  *  initial  velocity  recoiling  parts 
v  =  velocity  of  combined  recoil 
t  =  corresponding  time  for  combined  recoil 
x  =  run  up  on  inclined  plane  to  combined  recoil 

We  have       wv_+w4700 

V  =  0.9  (— )  (!•) 

wr 

wr  ,V-vcos(0+9), 
P-wrsin0  =  —  [ ]         (21) 

«         „   "r  v  sin(0+9) 

N  -  wr  cos  0  =  —  s (31) 

g      t    w 
P  cos(0+9)-Nsin(0+9)-wrsin  9-R  =  —  -    (4') 

g  t 
v      *r+we 

••st'-sT1"  -\  «**»*.$&«  (6>> 

_  v 

-  2  ^6  ' 

Usually  Z  and  x  are  given.  Hence,  the  unknowns 
are  V,  P,  t,  v,  K  and  R;  therefore  a  complete 
solution  is  possible. 

A  final  check  may  be  made  by  substitution  in 
the  energy  equation: 

PZ  -wr(Z  sin0-x  sin  9)+wcx  sin  9  +  i(mr+mc)v*+Rxi 
=  ^mrVa  (71) 

where  xt  =  -  t   or  in  the  form 
PZ-wr(Zsin0-x  sin  9)+wcx  sin  9  +Rx  =  7"irV* 

In  a  preliminary  layout  for  a  double  recoil 
system,  the  limitations  are  usually  the  length 
of  upper  recoil,  that  is  the  total  relative  dis- 


766 


placement  between  the  gun  and  top  carriage,  and 
the  total  run  up  the  inclined  plane.   A  direct 
solution  of  the  various  reactions  in  terms  of 
these  given  quantities  is  especially  useful. 
a  a 


h  *  cos(0+9) 

b  =  M 
g 

1  =  sin(0+6) 

nrcos(0+9) 

c  =  - 

g 

n  =  i»r  sin  9 
d  *  nrcos2f 

"c 

g  *  — 
g 

wrsin(0+6) 
f  «  - 
g 


2« 

2Z 

(6)  t  =  —  same  as  (5)  gives  t  directly 

b     v 

(7)  p  »  a+-  -  c  -       f(p.V)  same  as  (1) 

t     t 

(8)  N  =  d  +  f  -         f(N.V)  same  as  (2) 

t 

(9)  hp  -  IN-  N-R  -  g-   f(p.N.R.V.)  same  as  (3) 


(10)    X  >  —  +  p  —       f(R.V)  same  as  (4) 
2      R 


Elimination  N 


(11)    hp  -  Id  -  i^  v  -  n  -  f  v  =  R   (8)  in  (9) 

t  t» 


767 


f  (P.R.7) 

pV2  ' — 

i-  5  v          from  10     f (R.V. ) 

o 

Elimination  R 


(12) 


If+g 
(13)      P  ,  --  *--  v  *  -_      2    n 


xh-  —v 

2     f(P.V) 


elimination  of  P 


b   cv   ld+n   Id  +  g 

~  •  -  •    * 


""-' 


,,_x  aht  bhx        bhV        chx   p        ch 

(15)  ahx v~t 2 T  T"       ~  x<dl+n) 

t(dl+n)y       x  fl+g 

+ (fl+g)V+  £-V2-pV*=0 

2 

ch  +   fl*g        _     aht        b_h        chx   _    t(ld-«-n)      ^   x(lf-t-g) 

2  2  2          2  t  2  t 

bhx 
+   ahx    +  -  x(ld+n)   =  0 

This   equation   may   be    put   in   the   form   a1 V2+b ' V+c '=0 
w_cos2(0+9)        wr+Wr        wrsina(0+6)        WP 

i    C  w 

a1  = +  +  — 

2g  2g        2g        2g 

wr   Wr+Wc   Wc 
or  a1  =  —  -  +  —   hence  a'  =  0 

2g     2g    2g 

Solution  of  b  ' 

aht  Z   " t 

+  =  w  sin  0  cos  (0  +  6)  -  =  sin(?  cos(0+8) 

2  V    V 


768 


bh   *rv          "rv 


2g 


"r 
003(0+8)  =  r^-  cos  (0+6) 

^ 


*  -  "  -  ;TT  -  *  TT"  cos2  (0+6) 

a        2gZ        2gZ 

0Qt       wrcos0sin(0+6)   2Z»r 

-  -    =  ~2Z  -  —  -  =  —  —  cos0sin(0+9) 


2V  V 

nt 


sin  9 
V         v 

xg  _  xVwc 
t     2gZ 

hence  xVwr   Zwr       wrV  WCZ     xVwc 

-  sin9  +  —  cos  (0+9) sin9+    - 

2gZ    V         2Wg  V       2gZ 

simplifying 

xV          Z  w  V 

(wr+Wp) sine  (wr+Wc)+-c-cos (0+9)  =»  b1 

2gZ         V  2g 

2g 


+  ahx  +  x*rsin0cos (0+9) 

bhx     wr^ 
+  =  x  — —  cos  (0+9) 

t      2gZ 

-  dlx  =  -  x  w  cos  0sin(0+9) 

-  xn  *  -  x  Wcsin  9 

xwrVa 

xwP[sin0cos(0+9)-  cos0sin(0+9)]+  cos(0+9)-xW.sin9 

2gZ 

-  xwr[8in(0+9)cos0-cos(0+e)sin0] 

-  xw.sin  9  +  cos (0+9)-  x  W.sin  9 

2gZ 


769 


xwrV2 
C1  =  — —  cos(0+9)  -  x  sin  9(wr+Wc) 

..V2 

cos(0+9)-  x  sin(wr+Wc) 


(wr+W,J(—  -  -  sin  9)+  cos(0+e) 

r   c   2gZ   V         2g 


As  an  example  of  the  solution  of  these  equations 
and  a  calculation  of  the  prime  reactions,  the  240 
m/m  Schneider  Howitzer  was  taken  with  the  top  car- 
riage moving  up  a  plane  inclined  at  6°  with  the 
horizontal  and  with  40"  upper  recoil  and  a  total 
of  30"  recoil  up  the  inclined  plane  for  the  lower 
recoil. 

Muzzle  velocity  Vn  1700ft/sec, 

Travel  up  the  plane x  30in. 

Length  of  Recoil L  40  in» 

Angle  of  elevation 0  20° 

Angle  of  plane  9  6° 

Weight  of  carriage Wc  11,500  Ibs, 

Weight  of  gun wr  15,800  Ibs, 

Weight  of  the  charge W  35  Ibs. 

Weight  of  projectile  w  356  Ibs. 

Relative  displacement  Z=L-  40  in. 

wVm+4700  w       ^356+1700+4700+35. 

V  «  0.9( )  =  0.9( ) 

wr  15800 

„  6070+1640       7710 

=  0.9  — — =  .9       =  44  ft/sec. 

158  158 

sin(0+9)=sin  26°  =  .4384 
cos(0+9)=cos  26°=. 981 
sin  9  =  sin  6°  =  .1045 
sin  0  =  sin  20°  =  .342 
cos  0  =  cos  20°  =  .9397 


770 


2,22   .1045).  ,    .891 

44  64.4 


=   27300C.5124  -,0081)+9620 
»   13770+9620 
b1    =   23,390 


, 
64.4x40 


.891  -  2.5   x   27300   x    .1045 


-   317406  -  7132 
c1    =   310,274 

23390  V  =  310274 


2Z   2x3.33 

t  =  —  =  — — —  =  .1515      from  (5) 
V       44 

13.2652           27300  2 

2.5  »  — x  .1515  +  -— — — -  from  (4) 

64. 4R  13-2652 
74459.408 

R 

74459.408 
R  =  =  49800 

1.4952 

13.2652x.4384 

N  »  15800  x  .891  +  490  x  — — - from  (2) 

•  1  bib 


=  14080  +  18810 
N  =  32890 

P  -  15800x,342  +  490  44~13'2652x -891   from 

.1515 

5404+490x212. 
P  -  109,500 
t  -  0.1515  ft/sec. 
V  »  13.2652 
R  »  49,800 
N  =  33,000          P  =  109,500 


771 


As  a  final  check  on  the  calculations,  the 
values  obtained  were  substituted  in  the  energy 
equation.   The  slight  discrepancy  between  the 
two  sides  of  the  equation  is  due  to  numerical 
approximation. 

v8  i 

PZ+  —  (Mr+Mc;+x  .Wcsin  e  +x'R=  -MrV+wr  (Zsin0-x  'sine  ) 

Ct 

x1  =   t  =  1.0048 


365000+74,  5000+1200+50,  000=  490,  700 
475,000+16,400        =  491,400 

dev.    1.42X 

EXACT  THEORY  FOR  CONSTANT     Let  x1  and  y1  =  the 
RESISTANCE  ON  BOTH  UPPER   coordinates  along  and 
AND  LOWER  RECOIL  SYSTEMS.   normal  to  the  axis  of 

the  bore  (upper  recoil 
coordinates  ). 
x  and  y  =  the  coordinates  along  and  normal  to  the 

inclined  plane  (lower  recoil  coordinates) 
mr  and  »r  =  mass  and  weight  of  recoiling  parts 

mc  and  wc  -  mass  and  weight  of  top  carriage  plus 

cradle 

v  =  velocity  of  any  instant  along  inclined  plane. 
?'  =  absolute  velocity  of  recoiling  mass  along  axis 

of  bore. 

t  =  time  from  beginning  of  recoil 
0  =  angle  of  elevation  of  gun 

9  =  angle  of  plane 

E  =  free  recoil  displacement  for  upper  recoiling 
parts  during  powder  period 

T  =  total  time  of  powder  pressure  period 
p  =  resistance  of  gun  recoil  system 


772 


N  =  upper  normal  reaction  between  recoiling  parts 

and  top  carriage 
R  -  lower  recoil  resistance  parallel  to  inclined 

plane 

n  =  the  coefficient  of  sliding  friction 
pb  =  total  powder  pressure  on  breech  at  instant  t, 
Hence  t1  =  time  of  common  recoil 

v   =  common  recoil  velocity  for  both  recoil- 

ing parts. 

xj  =  absolute  displacement  in  the  direction 
of  the  bore  to  where  the  recoiling 
masses  move  with  common  velocity 
x   =  corresponding  displacement  up  inclined 

plane  at  common  velocity 
Z  =  total  relative  displacement  between 

upper  and  lower  recoiling  mass. 
Vf  =  free  velocity  of  recoil  (See  "Dynamics 

of  Recoil") 
RlJ  =  counter  recoil  buffer  resistance  for 

upper  recoil  system. 

Considering  now,  the  motion  of  the  upper  recoil- 
ing parts,  we  have  , 

Pb-P+wrsin0  =mr  -    (1) 
dt 

dv 
N  =  wrcos  0=mr  —  sin  (0+8)-  (2) 

From  (1),  we  have 


that  is, 


t  Pfcdt   (p-wrsin6) 
/   ---  t 


(p-w_sin0) 
V.  --  -  -  t  =  v'  (3) 

mr 

now,  when  t  =  T  V^=  Vj  •  hence  at  any  time  after 
T,  we  have 

(p-wrsin0) 
Vf  i  --  t  =  v  ' 


773 


Integrating  again  for  the  upper  recoil  absolute 
displacement,  we  have 

t        (p-w_sin0) 
Vfdt  - 


0          2mr 


but          tit 

/       Vfdt     =     /       Vfdt     +     /       Vfidt 
o  o  T 

Hence  (P-w_sin0) 

x'    =  E   +  Vft(t-T) t»  (4) 

2»r 

which  gives  the  absolute  displacement  of  the  upper 
recoiling  parts  along  the  axis  of  the  bore. 

Considering  now,  the  motion  of  the  lower  re- 
coiling parts,  ne  have, 

p  cos  (0+6)  -  JT  sin (0+8)  -  wcsin  e-R»«  —   (5) 

'  dt 

Substituting  N  from  equation  (2)  into  (5)  and 
simplifying,  we  have 

p  cos(0+9>-wrsin(0+9)cos0-wcsin  9-R=[flic-mrsin*00+e)~ 

dt 

(6) 

p  cos (0+9 )-wrsin (0+9 )cos0-wcsin9-R 

Hence  v  =  [ ]  t   (7) 

mc-mrsin*(0+9) 

and  the  corresponding  displacement  up  the  plane, 
becomes 

pcos(0+9)-wrsin(0+9)cos0-wcsin0-R 
x  -  t = ]  t.     (8) 

2[mc-mrsin»(0+9)] 

The  relative  velocity  between  the  upper  and 
lower  recoiling  parts,  become  vr=v'-v  cos(0+9)   (9) 
and  the  corresponding  relative  displacement 
xp=x'-x  cos  (0+9)  (10) 

When  the  upper  and  lower  recoiling  mass  move  to- 
gether with  a  common  velocity,  vr  =  0,  hence 
v'«v  cos (0+9)  hence  we  obtain  the  time  t1  for  the 


774 


comaon  velocity,  from 

p-w_sin0      pcos(0+9)-w_sin(0+9)cos0-wftsin9-B 
?ff  ,  (  -  £  -  )  t*  [ 


nc-«rsin*(0+9) 


cos  (0+9)  t1 
simplified,  we  have 


p-wrsin0    Pcos(0+9)-wrsin(09)cos0-wr3in9-R 
«r  mc-mrsin»(0+9) 


(12) 


cos (0+9) 

As  a  check,  the  time  t1  for  attaining  the  common 
velocity  of  the  upper  and  loner  recoil  masses, 
we  may  equate  the  components  of  the  absolute 
velocities  of  the  upper  and  loner  recoiling  mass 
parallel  to  the  inclined  plane. 

Considering  the  motion  of  the  recoiling  parts 
parallel  to  the  inclined  plane,  we  have 

v  »  Vf •  cos(0+6)  -  £-  cos(0+6)t'+  —  sin(0+8)tf 
«r  »r 


since  now  the  reaction  N  has  a  component  N 
reacting  on  the  upper  recoiling  parts  parallel  to 
the  inclined  plane. 

Let  Nt*  =  wrcos  0  t1  =  •rsin(0+9)  v  hence 

D  "r 

+9)-  -2-  cos(0+6)t'+  —  sin(0+6  )cos0t  '+sin* 

•r  "r 

"r 
(ef+9)  v  --  sin  6  =  v 

• 


0+e)  =  Vf,(0+6)-  S-  cos(0+6)  t'  +  ~ 
mr  mp 

cos0-sin  9]  t 
Let  sin(0+9)  cos(?-sin  6=(sin0cos0+cose)cos0-sin9 


775 


*  sin£fcos0cos9  +  cos*  0  sin  9  -  sin  9 

=  sin2fcos0cos6  +  sin  e  (cos'0-l) 

=  sin0cos0  cos9  -  sin  9  sin*  0 

=  sin0cos(0+9) 

bence  p-wrsin0 

v  cos(0+9)=V£i  -  (        )t ' 


r 


Substituting  for  v  and  reducing,  we  have,  as  before 

t,  _  !£J  _ 

p-wrsin0   p6os  (0+9  )-wrsin  (0+9  )-wcsin8-R 


.  -,..   . 
Br  mc-fflrsin«(0+9) 

Knowing  the  value  of  t'  and  substituting  in  equations 
(4)  and  (8),  we  obtain  the  total  relative  displace- 
ment between  the  upper  and  lower  recoiling  parts, 
that  is  Z  =  xj-  xtcos(0+9)  (13) 

where  t1  is  used  in  the  values  of  x1  and  x  res- 
pectively. 

The  total  energy  of  the  system  where  the  two 
masses  arrive  at  the  common  velocity  vt,  becomes 

Z  Z 

-(rar+Rc)v*  +  /   Pa^Z   where  /  padg  is  the  potential 

energy  in  the 

recuperator.   Let  R^  =  the  buffer  resistance  during 

counter  recoil  for  the  upper 
recoil  system,  then 

Z 
/  RjJ  d  Z  =  the  work  done  by  the  buffer  in  the 

upper  recoil  system. 
If  now  we  assume  that  the  counter  recoil  of  the 

upper  system  is  completed  during  the  recoil  of  the 
lower  system,  we  have 

Z  Z 

i(mr+mc)v;  +  /  p  d  Z  =  B(X-xt)+  /   R,J  d  Z 

=  wr(X-xt)sin0+wrZsin0+wc(X-xi)sin  9      (14) 
A  physical  meaning  and  relationship  of  the  re- 
actions in  this  equation,  may  be  had  oy  a  con- 
sideration of  the  component  dynamical  equations 
for  the  parts'  of  the  system. 


776 


Daring  this  second  period  of  the  recoil, 
we  have  for  the  lower  recoiling  mass,  that 
[R+wcsine+Nsin(0+e)-(pa-Rb')cos(0+e)]dx  =  -  ncdV 
and  for  the  upper  recoiling  parts,  along  and 
normal  to  the  boref  (pa~RjJ  )-wrsin0)d(x  cos(0+9)+Z] 
~mrvxdvx  and  (wreos0-H)d[x  sin(0+e)}*-mrvjdvj 
and  the  above  equations  and  integrating  the  sum, 
ire  have 


X          X 

/   R  dx 

»a 


+  /   wrsin6  dx  +  /   (pa-Rb«  )dZ+wr  f  [sin 
*  t  x 


(0+6)  cos  0  -  cos  (0  +  6)sin0]  dx 
o  v« 

-  /  wrsin0  dZ=(mr+mc)r^  (since  v£*+  v^2=  v*  for 


initial  valve) 
Simplifying  we  obtain  equation  (14) 

In  general  the  potential  energy  of  the  re- 
cuperator is  partially  divided  in  overcoming  the 
work  of  the  upper  recoil  buffer 

Z 
/  R£  dZ  and  in  augmenting  the  run  up  the  inclined 

plane  over  that  if  there  were  no  re- 
cuperator present.   Hence  in  general 

Z         Z  ^  2 

/   padZ  >  /  Rfa'  dZ  and  /   p.  dZ  -  /   RJ  dZ 
o         o  o  o 

is  the  additional  2nergy  over  that  Kinetic  energy 
at  common  velocity  which  augments  the  recoil  up 
the  inclined  plane. 

We  nay  assume  with  small  error,  however,  that 

Z          Z  Z          Z 

/  pa  dZ  -  /  RjJ  dZ  or  that  /  pa  dZ  -  /  R^  dZ  is 

negligible.   This  does  not  imply  that  pa-R£=  0. 
Since  (pa~R^ ) (X-xt)  cos (0+6)  (roughly)  is  the 

agent  by  which  the  upper  recoil  energy  is  dissipated, 
When  the  lower  recoil  is  comparatively  short 


777 


and  the  resistance  of  the  lower  recoil  system  R 
is  large,  we  have  often  a  condition,  where 
counter  recoil  in  the  upper  recoil  system  be- 
comes impossible  and  we  even  have  an  over  run  of 
the  upper  recoil  system. 

Thus  assuming  during  the  second  part  of 
double  recoil,  that  the  upper  and  lower  recoil 
mass  move  as  if  one,  we  have  the  retardation 

dv   R+  (wr+wc)sine 

-  —  -  *  -——_———   and  for  the  upper  recoil- 
df    mr+mc 

ing  mass 

dv 

Pa-  wr  sin  0=  mr(  --  )eos(0+e)    hence 
dt 


m 


Pa-wrsin0  =  [R+(wr+wc)sin9]cos(0+9) 

mr+mc 

Now  if  pa  >  wr  sin  0  +  — —  [R  +  (wr-rw_)sin0]cos(0+9) 

mp+mc 

Counter  recoil  of  the  upper  recoil  system  is  pos- 
sible during  the  second  period  of  the  lower  recoil 
system.   If  however, 

mr 
Pa  <  wrsin0+  [R+(wr+wc)sin  9]eos(0+6) 

We  have  a  tendency  of  over  recoil  of  the  upper  re- 
coil system  hence  counter  recoil  of  the  upper  re- 
coil system  is  impossible.  For  this  case  the  energy 
equation  reduces  with  exactress  to 

i(mr+mc)v«*H(X-xt)+(*r+wc)(I-xt)sin  6    (15) 

The  velocity  curve  during  the  second  period 
may  be  obtained  with  sufficient  exactness  by  as- 
suming the  two  masses  to  recoil  together,  then 

<*» 
R+(wr+wc)sin8a-(mr+mc)v  — 

x  v 

and  /        [R+(wr+wc)sine)dJc  »   )          (mr+rac)v     dv 


778 


mr+mc 
[R+(wr+wc)sin6]  (x-xt)  =  (-— )(va-v* 

hence      /~   a[8*(.r*n0)«ine]U-«t) 

?.»/.?• ' 

mr+mc 

,   RECAPITULATION  OP  FORMULAE     Prom  approximate 
FOR  CONSTANT  RESISTANCE  TO   solution  with  limited 
RECOIL  BOTH  UPPER  AND       upper  and  lower  re- 
LOWER.  coil,  calculate 

P  and  R.   Then  during 
the  powder  period,  we  have 

p-wrsin9 

v'  .  Vf  ~  ( )  t 

rar 

pcos(0+9)-wrsin(0+6)co30-wcsin9-R 
mc-mrsin2(0+9) 

and  the  relative  velocity  becomes  v^v'-v  cos(0+9) 

t           p-w_sin0 
x1  =  /   Vf  idt  -  ( )t« 

o  2mr 

pcos(0+9)-wrsin(0+9)cos£l-wcsin  6-R 


and  the  corresponding  relative  displacement,  be- 
comes xr=x  '  -  x  cos(0+6).   After  the  powder  period 

during  the  remainder  of  the  first  period  of  recoil, 

we  have        p-wrsine! 

v'=V  i(  -  )t 


3  t 


and  for  the  relative  velocity  vr»v'-v  cos(£l+9) 

Furtber  p-w.sinar 

x'«E+Vf i(t-T)  -  ( )tf 

2m_ 


779 


p  cos(0+9)-w  sin(0+9)cos0-w.sin9-R 
x  .[  -  £  -  2  -  ]t. 

2ac-»rsina(0+9) 

and  the  corresponding  relative  displacement  be- 
comes xr»x'-x  cos  (0+8).  The  tine  for  the  common 
velocity  becomes, 

Vf, 


t'= 


p-wpsin0   pcos  (0+9  )-wrsin  (0+6  )cos0-wcsin9-R 

.,  fl.  -  ]cos(0+9) 


and  the  common  velocity  becomes 

pcos  (0+9  )-Wpsin  (0+9  )cos0-wcsin9-R 

V**t-      mc-H.rsin»(0+9)          -U' 

p-wpsin0 
x'=E+Vf  ,(t'-T)-(—  -  -  )t' 

-oDDj. 

pcos  (0+9  )-w_sin  (0+9  )cos0-w_sin  6-R 
x  -f  -  -  -  ]ta 
2[mc-mrsin«  (0+9)] 

and  the  total  relative  displacement  for  the  upper 
recoil  system  becomes,  Z  *  xj  -  xtcos(0+9) 

Oaring  the  second  period  of  the  recoil,  ire 
have 

v*-2(R+wr+wc)sin  9(x-xt) 

v  =  i 

mr+mc 

the  upper  recoiling  mass  being  assumed  locked  with 
the  lower  recoiling  parts. 

CALCULATION  OF  THROTTLING  GROOVES     As  a  first 
BOTH  UPPER  AND  LOWER  RECOIL.        approximation, 

it  will  be 
assumed  that 
the  total 

friction  is  mainly  guide  friction  and  proportional 
to  the  normal  reaction  between  the  upper  and  lower 
recoiling  parts.  Then  Rg*  nN  where  n  *  0.2  to  0.3 

and  N  »  w_cos0  +mp  —  sin(0+9)  where  v  and  t1  have 
t' 


780 


been  already  determined.  Considering  the  upper 

recoiling  parts,  we  have  P»pa+pn+Rg*pa+p_+nN 
hence  pn=p-pa-nN.  Further  if  the  ratio  of  the 

final  to  the  initial  air  pressure  =  m,  we  have 

Paf 

— —  «  m     and  if  A  «  effective  area  of  upper 

°ai  recuperator  and  b  =  Z  = 

recoil  displacement  on  top  carriage,  then  for  the 

initial  volume  we  have 

x 

7j  *  Aab     •    where  k  =  1  to  1.41  assume  1.3 

mk  -  1 

vi 
Hence  pa=Pai(       ) 

vi  *axr 

xr  =  being  the  relative  displacement.   There- 
fore knowing  vr  and  xr  and  the  total  pull  p,  we 
have 


/   -i   \  »» "n  *  r 

PnsP-PaiV-A    >  and  Wx i    = — 


Vj_k-nN  KAnV 

i-Aa 

13. ay 

Ah 

where  An=  the  hydraulic  piston  area  and  k  the 
reciprocal  of  the  throttling  constant. 
Lower  throttling  grooves 

Knowing  R  from  previous  data,  we  have 

R  A  v 

where  v  is  the  recoil  velocity  up 


13. 2/-   plane. 


EQUIVALENT  MASS  OP  ROTATING     When  a  double  re- 
PARTS  WITH  A  DOUBLE  RECOIL.   coil  system,  con- 
sisting of  two 
separate  recoil 
systems  is  used, 

mounted  on  a  railway  car  or  caterpillar,  it  is 
customary  to  consider  the  car  or  caterpillar 


781 


sufficiently  braked  to  allow  no  recoil.   In  fact 
a  salient  feature  of  the  design  is  to  make  "R*  small 
enough  so  that  the  rail  or  ground  friction,  induced 
by  proper  braking,  is  sufficient  to  balance  R. 

Due  to  the  complication  of  a  double  recoil, 
as  well  as  the  impossibility  with  very  large  mounts 
of  taking  up  the  recoil  energy  even  with  a  double 
recoil  system  without  an  excessive  recoil  displace- 
ment it  has  been  the  custom  to  use  a  single  recoil 
and  allow  the  railway  car  or  caterpillar  to  run 
back  a  limited  distance  dependent  on  the  magnitude 
of  the  braking.   The  recoil  of  the  car  on  very  large 
railway  mounts  may  be  considerable.  This  greatly 
reduces  the  stresses  on  low  elevation  as  well  as 
augments  the  stability.   In  fact  with  such  mounts 
stability  is  of  no  longer  a  consideration. 

When  a  single  recoil  system  is  used  but  the 
car  or  caterpillar  recoils  in  addition,  we  obvious- 
ly have  a  double  recoil  system  and  all  the  prevous 
dynamical  equations  together  with  the  method  of 
computing  the  throttling  on  the  upper  recoil  or 
now  the  recoil  systems  holds  the  same.  The 
lower  recoil  resistance  R  is  now  the  tangential 
reaction  exerted  at  the  base  of  the  car  wheels  or 
at  contact  of  ground  and  caterpillar  track.   In 
the  acceleration  of  a  railway  train,  railway  engineers 
customarily  allow  for  the  rotational  inertia  of  the 
car  wheels  by  increasing  the  translatory  mass  from 
8  to  10  percent.   Due  to  the  limitations  at  times 
of  car  or  caterpillar  recoil  and  the  great 
variation  of  the  magnitude  of  the  rotational  in- 
ertia as  compared  with  standard  railway  practice 
it  is  important  to  calculate  the  exact  effect  of 
the  rotational  inertia  in  terms  of  an  equivalent 
addition  to  the  translatory  mass. 

Consider  a  railway  car  or  truck  with  "n" 
pairs  of  axles.   Let 

wc  and  mc  =  weight  and  mass  of  car  not  in- 


782 


eluding  wheels. 

•„  and  BW  *  weight  and  mass  of  a  pair  of  wheels. 
I  *  D*K*  =  moment  of  inertia  of  a  pair  of 

wheels  about  the  center  line  of  the 

axle. 
d  *  tread  dies.  of  a  car  wheel 

k  *  radius  of  gyration  of  a  pair  of  car  wheels 
Nw  =  normal  reaction  at  base  of  a  pair  of 

car  wheel  x 
Kg  *  normal  reaction  of  brake  shoe  on  wheel 

per  pair  of  wheels 
fw  «  coefficient  of  rail  friction 
fa  *  coefficient  of  brake  shoe  friction 
R,,  *  tangential  force  exerted  by  rail  oc  base 

of  car  wheel 
p  *  recoil  reaction 
N  *  normal  reaction  between  recoiling  parts 

and  car 

0  *  angle  of  elevation 

Now  independent  of  rotation  or  any  other 
motion,  the  translatory  motion  of  the  center  of 
gravity  of  a  system  depends  only  on  the  external 
forces  applied.   Hence 

p  cos  0  -  N  sin  £>  -  2  Rw  «  (mc+Znw)  — 

dt 
Considering  the  motion  of  a  single  car  wheel,  we 

have  for  rotations  about  the  center  of  gravity 
of  a  pair  of  wheels 


hence  pcos0-Nsin0-2N8f  s=(mc*2mlf+Z  —  -  —  ;—  there- 
fore the  translatory  mass  is  increased  by  the 

4mk« 

which  is  the  equivalent  translatory 


783 


mass  of  rotational  inertia.   The  a  ass  of  the 
lower  recoil  system  therefore,  becomes 

4k* 

m-+2m_(l+— -—)   and  this  value  is  to  be  sub- 
0       d 

stituted  in  the  previous  dynamic 

equations.   The  equivalent  resistance  for  R  is 
now  the  summation  of  the  brake  shoe  friction,-  that 
is  R  =  Z  Ngfg  and  this  value  is  to  be  used  in  place 
of  R  in  the  previous  dynamic  equations. 

It  is  important  to  note,  however,  that  the 
actual  tangential  force  exerted  at  the  base  of  the 

*   :  ..  • 

car  wheels  is  not  2Nsfs 

but  ZR,  -  2Ngfs  +  2  — JT-  - 

943  '-is  '-'-si*  •          ->riJ-afr  HoJ«H»i  srff 
Consider  a  caterpillar  track  and  connector 

mechanism: 


Let  Rt»  total  tangential  track  reaction  between 

track  and  ground  (in  Ibs) 
Rw  =  total  tangential  roller  reaction  on 

track  (in  Ibs) 

rw  =  radius  of  roller  wheel  (in  ft) 
rc  *  radius  of  sprocket  (in  ft) 
rt  *  radius  of  sprocket  gear  (in  ft) 
r  =  radius  of  brake  drum  gear  (in  ft) 
r  =  radius  of  drum  of  brake  clutch  (in  ft) 
R  *  tangential  reaction  between  sprocket 

gear  and  drum  gear  (in  ft) 
TQ  =  torque  exerted  on  brake  drum  (Ib.ft) 
E0  =  mechanical  efficiency  of  sprocket 

mechanism 
Et  *  mechanical  efficiency  of  transmission 

between  sprocket  gear  and  drum  gear. 
Efi  -  mechanical  efficiency  of  brake  drum 

and  mechanism 

EN  3  mechanical  efficiency  of  roller  trucks. 
A  =  resultant  normal  bearing  reaction  of 

sprocket  shaft  (Ibs) 
J*  »  resultant  normal  bearing  reaction  of 


784 


brake  drum  shaft  (Ibs) 

ft  and  ff  «  corresponding  coefficients  of  friction 

BC  =  total  mass  of  caterpillar  excluding  recoiling 

parts 

mwk£  =  moment  of  inertia  of  roller  wheel  (ft. Ibs) 
msk|  =  moment  of  inertia  of  sprocket  wheel 
mgsk§s  *  moment  of  inertia  of  sprocket  gear 
=  moment  of  inertia  of  drum  gear 
1  Moment  of  inertia  of  drum, 
d  *  the  increment  change  in  the  radius  to  account 
for  friction  between  gear  teeth 

Considering  the  motion  of  the  caterpillar, 

we  have 

dv 

p  cos  0-N  sin  0  -Rt  *  mc  77 

The  tension  in  the  caterpillar  track  at  the 
sprocket  becomes,  T  =  Rt-2Rw 

.  R    *"k"   d*    (2) 
Bwr*   dt 

and  its  moment  about  the  sprocket  axis,  (assum- 
ing the  upper  track  tension  as  nil)  becomes, 


(Rt 


mwkw 


«  dt 


Considering  the  angular  motion  of  the  sprocket  shaft 
we  have 

mwkw  dv  ,  mskg-nng3Kgs  dy 

' 


(3) 

Further  considering  the  angular  motion  of  the  drum 
shaft,  we  have 

dv 

(0 


hence 

si,   „, 


785 


Where  E^  takes  care  of  the  friction  less  in  the 
drum  of  gear  bearing. 

The  friction  loss  between  the  drum  gear  and 
sprocket  gear  may  be  considered,  by  letting 


Illl  *  1  li 
r,-d  3  E  r, 

hence 


Substituting  in  (3),  we  have 

v£  dv   TQ  ix 

Rt     »    ^o- 


EBr»      dtoE  EEErr»t   dt 


where  EO  takes  care  of  the  friction  loss  in  the 
sprocket  bearing.   Therefore,  the  track  re- 
action becomes, 

k|        dy 


(8) 


and  substituting  in  the  equation  of  translatory 
motion  (Eq.l)  we  have 

TD    rt        mHk« 

p  cos  0-  N  sin0 — —  =  [mc+  '.""    + 

BoM.  r.r0       EBrS 


^ 


TD 

Evidently  —  —  —  —  is  the  brake  torque  referred 
EoEtE2r2ro  to  a  reaction  at  the  base  of 
the  track,  considering  the  mechanical  efficiency 
of  the  gearing  .   The  translatory  mass  is  augmented 


786 


due  to  the  rotational  inertia  of  the  rotating 
parts  by  the  terra 

2   msks*mgsk|s 


which  is  the  equivalent  mass  of  the  rotating  ele- 
ments. 

It  is  to  be  noted  that  the  mechanical  ef- 
ficiency enters  in  the  rotational  inertia  since 
the  bearing  reactions  depend  upon  the  external 
reactions,  and  the  moments  of  them  in  turn  de- 
pend upon  the  rotational  as  well  as  translatory 
inertia.   The  effect  of  the  translatory  inertia 
on  the  rotating  element  in  modifying  the  bearing 
reactions  will  be  neglected,  being  small. 

Hence  R  in  the  double  recoil  equations  is  now 
the  braking  torque  referred  as  a  tangential  force 
at  the  track  base,  that  is, 


The  actual  tr  ck  re.action  is  Rt  given  by 
equation  (8).   As  a  check  on  equation  (9)  ire  may 

note  that  from  the  energy  equation,  we  have 

t      t 

D  9  '     ^dda^nifi^fid    ft    i 
p  cos0-N  sin0  dx  =  »-d  (  )w'  +d(5-racv*) 


the  reaction  Rt  doing  no  work.   Further,  we  have 

r,  ri  v 

de*  %-r4-  dx  K  =  r  r 

«  O  a  0 


V  V 

If  »  W_  =  

r  "    r 

ro  rw 

hence  substituting  these  values,  we  find 


787 


T  r 


.  :t.0 


therefore  the  equivalent  translatory  mass,  to  ac 
count  for  the  rotational  inertia  becomes, 


gdk|d   a   msks+  mgsk|s 
' 


When  the  caterpillar  track  is  heavy  or  there 
is  a  long  space  between  the  driving  sprocket 
and  the  front  idler  sprocket,  its  inertia  ef- 
fect must  be  considered.   Therefore,  let 

r0  =  radius  of  drive  gear  sprocket  (in  ft) 
r  =  radius  of  front  idler  sprocket  (in  ft) 

t  fc  »  «  i    i  " 

m^k£  =  moment  of  inertia  of  idler  sprocket 

(units  Ib.ft) 

mt  =  mass  of  caterpillar  track  per  unit 
length 

1  *  length  of  upper  span  of  caterpillar  track 
(in  ft) 

(aromt)r£  =  moment  of  inertia  for  that  part 
of  track  in  contact  with  driving 
sprocket  (units  in  Ib.  ft) 

(nr^mt)r£  =  moment  of  inertia  for  that  part 
of  track  in  contact  with  front 
idler  sprocket  (units  Ib.  ft.) 

T0  =  tension  at  section  at  point  of  contact 

of  lower  track  and  drive  sprocket  wheel 
(in  Ibs) 
T.  =  tension  at  point  of  contact  of  lower 

3 

track  and  front  idler  sprocket  (in  Ibs) 
Ta  =  tension  at  point  of  contact  of  upper 

track  and  idler  sprocket 
T  =  tension  at  point  of  contact  of  upper 

track  and  drive  sprocket 
From  kinematics  we  must  have  the  relative 
velocity  of  the  track  with  respect  to  the  frame 


788 


dv 
equal  to  v  and  the  corresponding  acceleration  — 

where  v  is  the  translatory  velocity  of  the 
caterpillar  at  instant  with  respect  to  the 
ground. 

Considering  the  lower  track  since  at  any 
instant  it  oust  be  at  rest,  we  have  for  the  dif- 
ference of  the  tensions  at  its  extremities, 

Zawkw  dv 
TO  -  T,  -  Rt  -  j-Jj!  g      (10) 

I***    at 

where  the  second  member  is  the  reaction  on  the 
track  due  to  the  tangential  reaction  of  the  ground 

and  the  reaction  of  the  truck  rollers  . 

Considering  the  angular  notion  of  the  drive 

sprocket  shaft,  we  have 


(To-Tt)rr-8t<rt+d)-Atftr'  •(  *  n  r 

(11) 
and  for  the  upper  track,  we  have 

T  -  T  =  2  ml  —  (12) 

1   *      *   dt 


•ii         dv 

(Tt-Ts)rt  »( — —  +  n  rjnt)—  +  A1firii-n  (13) 
ri         dt 

From  equations  (4), (5)  and  (6)  in  the  pre- 
ceding discussion,  combining  with  (11),  we  have, 

s?sktfs  dv 

gs  gs  ^  n     }]dl 

dt 


(14) 


Further  equation  (13)  may  be  simplified  by 
considering  the  mechanical  efficiency  of  the 
front  idler  sprocket  mechanism  Ej,  that  is 


789 


<T.-IK  •<-ir'  +  n  r*  mt)IT  Ei 

where  E  takes  care  of  the  bearing  friction  A^£^ 

and  the  loss  due  to  the  Deeding  of  the  track  at 
the  sprocket. 

Now  if  we  combine  (10),  (12)  and  (15)  with 
(14),  we  have 

Zmwkw        ,  nri   ?rox  miki  c   /gd^gd, 
D   =  -  +  m+  (2—  1+  -  +  -  —  )•*—  —  —  E^  +  (-?..  8  ..    )r2 
*   Ewr*     ^    Ex    E0    rj   «  Vl         * 

kj  s  dv     TD    rt 


E0r§ 


nri   nro 

but  mt(2  1  +  -—  +  - — )=Mt  approx.  (total  mass  of 
Bi    Eo  track) 

dv     TD    ri 
B*=Me  dt  +  ^E^   ral.o 

Substituting  in  the  equation  of  translatory  motion 
we  have, 

TD     ri          dv 

p  cos-  N  sintf =(Kc*Me)—     (17) 

Eo£tE2  raro        dt 

Where  Me  given  in  equation  (16)  is  the  equivalent 
mass  that  must  be  added  to  the  translatory  mass. 

The  equivalent  inertia  may  be  taken  into 
consideration  more  simply  by  tlie  following  ap- 
proximate method. 

The  primary  rotational  system,  consists  of 
the  track,  the  drive  sprocket  and  front  idler, 
together  with  the  truck  rollers. 

The  reaction  of  the  ground  tangentially  to 
the  track, =  Rt  and  the  truck  roller  reaction  = 

mw'lw  dv 

~  at 

Hence,  for  tbe  primary  rotational  system,  we  have, 


790 


m»kw 


r0)—  (18) 

but  considering  the  rotational  system  of  the  drum 
and  gear,  we  have 


Combining  (18)  and  (19)  we  have 


Hence,  we  assume  the  radius  of  the  idler  sprocket 
and  driven  sprocket  the  same,  namely,  rQ 

To  account  for  the  loss  due  to  friction,  let 
the  Mechanical  efficiency  of  the  greasing  re- 
ferred to  the  track,  be  as  follows: 

Et  =  mechanical  efficiency  of  track 

EI  =  J4.E.  of  front  idler 

Eo  =  M.E.  of  drive  sprocket 

Et  =  M.E.  of  gear  transmission 

Ef  =  M.E.  of  druu  shaft 

Ew=  M.E.  of  truck  rollers 
Then  equation  (20)  is  modified  to: 


V, 
=U  -  (21) 


tbat  "      d»     TO', 
Rt  "  *c 

d  t      E>n^>  _  E_  1 


PRIMARY  EXTERNAL  REACTIONS  WITH     With  a  double 
A  DOUBLE  RECOIL  SYSTEM.          recoil  system, the 


791 


first  period  when 

the  top  carriage  is 
accelerated  to  a 

common  velocity  for  both  upper  and  lower  re- 
coiling parts  and  a  second  period  with  a  re- 
tardation for  both  recoiling  masses. 

The  reactions  should  be  considered  during 
both  periods. 
External  reactions  during  first  period: 

By  O'Alembert's  principle  we  may  regard  the 
inertia  force  as  an  equilibrating  force,  then  for 
the  primary  external  forces  of  a  system  con- 
sisting of  the  upper  and  lower  recoiling  mass  to- 
gether with  caterpillar  or  railway  car. 

(1)  The  inertia  resistance  of  the 
recoiling  mass  divided  into  two 

components . 

(a)  The  inertia  force  parallel 
to  axis  of  the  tore  through 
the  center  of  gravity  of  the 

upper  recoiling  parts, p1  or  K 

xt 

(b)  The  inertia  force  normal 

to  the  upper  guides  through  the 
center  of  gravity  of  upper  re- 
coiling parts,  N1  or  Kv 

*\ 

(2)  The  weight  of  the  recoiling  mass 
acting  vertically  down  =  Vr 

(3)  The  inertia  resistance  of  the 
top  carriage  and  cradle  acting 
through  the  center  of  gravity  of 
the  top  carriage  and  cradle 
parallel  to  the  inclined  plane 
opposite  to  the  acceleration  up  the 

plane  =  Kx  or  mc  &— * 
dt  * 

(4)  The  total  weight  of  the  top 
carriage  and  cradle  acting  vertically 


792 


down  =  «c 

(5)     The  reaction  of  the  ground  on 
the  caterpillar  track  or  the  re- 
action of  the  rail  on.  the  braked 
wheels  of  a  railway  mount  using  a 
double  recoil,  which  are  divided 

into  the  following  components: 
(a)     The  tangential  reaction 

of  ground  or  rail. 
(b  )     The  normal  reaction  of 
ground  or  rail  which  is  not 
uniform  but  distributed  so  as 
to  produce  an  upward  normal 
reaction  combined  with  a  couple. 
When  the  mount  is  just  stable  as  with  a  light 
caterpillar  at  zero  elevation  (5)  reduces  to  a 
single  reaction  about  which  moments  are  taken  and 
therefore  would  not  be  considered  for  critical 
stability. 

The  primary  external  reactions  are  shown  in 
fig.  (3). 

Considering  the  motion  of  the  upper  recoiling 
parts,  we  have,  during  the  powder  period, 


Pb-P+Wrsin  0  =mr  —  —   for  the  acceleration 

of  the  upper  recoil- 
ing parts  parallel  to  the  guides.    And 


A 
N  -  Wrcos0  =  mr  -  -  =  mr  —  sin  (0+9)   for  the  ac- 

dt*       dt          celeration 
of  the  upper  recoiling  parts  normal  to  the  upper 
guides. 

The  external  reaction  on  the  recoiling  parts 
when  considered  with  the  total  mount,  becomes, 

during  the  powder  period, 

d2x 
P'=KX  *Pb-mr  -  -=P-W-sin£! 

r«     dt« 

parallel  to  the  guides  in  the  direction  of  Pb. 

After  the  powder  period  during  the  first  period 
of  recoil,  we  have 


793 


P-Wrsin0=-mr along  the  guides  and  H-Wrcos0  - 

dt* 
dv 
mr  —  sin(0+9)  normal  to  the  guides.   Therefore 

the  external  forces  on  the  re- 
coiling parts  during  the  first  period  after  the 
powder  period,  becomes. 

P'=KX  =-m reversed  =m_ =P-Jf,sin0  along  the 

1   dt*  «•          bore  and 

dv 
tf'=Kg  =mrI7  sin(0+6)  =  N-Wrcos0  normal  to  the  bore. 

Hence  during  the  first  period  of  recoil  either 
during  or  after  the  powder  period,  we  have 
P'=KX  =P-Wrsin0  along  the  bore  downward  and 

1   dv 

N'  =  KV  =rar™  sin(0+6)  normal  to  the  bore  dowaward 
*  t   d  t 

or  =  N-Wrcos2f.  We  have  in  the  above  neglected  the 
powder  pressure  couple, it  being  at  best  small,  witb 
little  or  no  effect  on  stability.   The  inertia 
force  of  the  top  carriage  is  evidently 

dv 

•C"T~  reversed  parallel  to  the  inclined  plane.   Hence 

the  external  forces  not  including  (5)  become 
(1)     pl=Ky  =P-Wrsin0  along  the  bore 

1   dv 

K'  +  KV  anir-—  sin(0+6)   normal  to  the  bore 
»t   dt 

(2)     Ur=  upper  recoiling  weight,  vertically 

down, 
d  v 

(3)  ""cJ"  =  inertia  force  of  lower  recoiling 

parts  parallel  to  inclined  plane. 

(4)  tfc  =  weight  of  lower  recoiling  parts, 

vertically  down. 
The  external  reactions  during  the  second  period: 

dv 

During  the  second  period,  mcT~  =  Kx  reverses 

d  t     t 

in  direction  since  the  lower  recoiling  mass  now 
becomes  retarded.   On  the  assumption  that  during 
this  period  the  upper  recoiling  parts  have  the 
same  notion  as  the  lower  recoiling  parts,  we  have 


794 


STABILITY 
COMPUTATION 


Y 


c*  Accf/tration    of  Carriage    up  jo/an* 


P  *  Broking  Force  -f=>-H/,.5"> 
t  •  Any/e  of  C/evo+ion 
6  •  flng/e  of  inc/ined  P/ane 
s  about  P 


r~or  any  of  her  point 
X'r  -  TCr  -ff'Cos 


6)      ,        Yr 


i.  3 


795 


DOUBLE  REDDL  SYSTEM 
STCHAMOND  &OMM.  HOWITZER  (SCHNEIDER, 


796 


for  the  inertia  force  of  the  upper  recoiling  parts: 

if 

mrr—  in  the  direction  up  the  plane  and  parallel  to 

the  inclined  plane.   Consideration  of  the 
forces  acting  when  counter  recoil  for  the  upper 
recoiling  parts  place  during  the  second  period  of 
recoil  will  be  taken  in  "variable  resistance  to 
recoil  for  the  upper  recoiling  parts". 
STABILITY  FOR  DOUBLE  RECOIL  SYSTEM.     Consider- 
ing fig. (3) 

let  ac=  **• 
c  dt 
=  ac- 
celeration of 

carriage  up  plane.   N'=niracsin(0+6) 

P'=P-Wrsin0  =  resistance  to  recoil  for  upper  re- 
coiling parts  parallel  to  guides. 

Wr  =  weight  of  recoiling  parts 

*t  =  weight  of  caterpillar. 

IYC  =  weight  of  lower  recoiling  parts 

Let,  xryr,  xcyc,  and  x^y^  be  the  respective  co- 
ordinates for  the  variops  weights,  from  the  over- 
turning point  0,  at  the  end  of  contact  of  the 
caterpillar  track  and  ground.   Then  for  moments 
about  0,  for  the  various  external  forces,  in 
battery,  we  have  MQ= (Nleos0+P'sin0+Wr)xr+ (wc+mcac 

sin6)xc  +  (Nlsin0-P'cos0)yr+(mcaccose)yc+wtxfc 

and  for  any  other  position  in  the  recoil,  the 
various  coordinates  of  the  above  equation  change 
to  xr=xr(3lcos0+3cos9),  yr=yr-(S  'sinfl-SsinS ) 

x£  =xc-Scos9,          y£=yc+Ssine 

•here  S'=/  vrel  dt,  and  3s/  vcdt 

vrel  =  relative  velocity  between  upper  and  lower 

recoiling  parts 

vc-  velocity  of  carriagrs  up  inclined  plane. 
Further  let  A  =  Wtxt+WrXr+WcX<! 
B=(N'cos0+P'sin0)xr 
C  =(N'sin0-P'cos0)yr 


797 


dvc 
D  =  (mc sin  6 )  xc 

dt 

dvc 

E  =  <mc~  cos  8>yc 

F=A+B+C+D+E 

For  stability,  we  jnust  have  F  =  0.  The  critical 
position  fir  stability  for  the  first  period  is 
at  the  end  of  the  first  period  when  the  two  re- 
coiling parts  begin  to  move  with  the  same 
velocity.   The  coordinates,  therefore,  become 
xr=xr-(Zcos0+xtcose  ),y^=yr-(Zsin0-xtsin  9) 

xcsxc~xicose>       vc=yc+xtsin9 

x  =dis placement  up  inclined  plane  at  common  velocity 
Z=total  relative  displacement  between  upper  and 
lower  recoiling  mass. 

Assuming  as  before  that  during  the  second 

period  the  two  recoiling  masses  move  as  if  one, 
we  have,  for  the  condition  of  stability 

Mo=Wtxt+wcxc+Wrxr~IDcacsin9  xc"  acaccos0  yc"mracsin 

P 
9  x£  -  mraccos  6  y'r  =0  where  ac= 

'  mc+mr 

and  the  critical  stability  is  at  the  end  of  recoil 
and  xr=xr-(Zcos0+Xcos9);  y '  =  yp  -(Zsintf-X  sin  e) 

xcaxc~Xcos  e>  yc=yc+x  sine 

where  X  is  the  total  run  up  the  plane. 

If,  however,  the  upper  recoiling  parts  move 
into  battery  during  the  second  period  while  the 
top  carriage  still  continues  moving  up  the  in- 
clined plane,  then  xr=xr-J[  cos9. 

STABILITY  WITH  A  SINGLE  RECOIL     We  have  as  be- 
AND  CATERPILLAR  BRAKED.        fore  the  same  in- 
ertia and  weight 
moments  but  in 
addition  rotational 

inertia  couples.   Since  the  effect  of  a  couple  is 
entirely  independent  of  the  axis  about  which  moments 
are  taken,  we  merely  have  to  add  in  the  previous 


798 


799 


moment  equation  the  additional  rotational  inertia 
terms,  taking  of  course  account  of  the  algebraic 
sign  of  the  inertia  couple. 

The  following  inertia  couples  are  intro- 
duced with  a  caterpillar  using  a  simple  mechanism 
as  assumed  before: 
Stabilizing  inertia  couples:— 


JT*  =  drive  sprocket  and  bear  couple 


dv 

—  =  roller  truck  couples 

dt 


mik!   dv 

—  —   —  =  front  idler  couple 

r      dt 


1  r]  =  track  inertia  moment  where  r  = 

ro+ri 

-   and  1  =  total  span  of  track 

2 
Overturning  inertia  couple:- 

mdk3+agdk|d   dv 

(  -  *~)*.T?  =  drum  shaft  inertia  couple 
r  r      *d"t 

Therefore  tfce  stability  equation 

becomes,  F  =  A+8+C+D+E+G+H+I+J+L  and  for  stability 
P  ^  0.  Where  during  the  first  period  A=ffrxt+Wrxr 

+W.XI   ,    B  =(N'cos(2f+PlsinCf)x'          C  *  (N'sin0-P'cos0)y  ' 
H  dv 

D  =    (fflc"^  sin  e>*c>        E  =    (mc~J7  cos 
skgskdv  "wkw  dv 


^  ^ 


ii  dv 


mk 

J  =»[n(r  +  r»)+2  1  r] 


/8ddx   dv 
-  (  -  'rt~"~      where  the  coordinates 

ror»     dt       refer  to  point  of 
contact  of  ground  and  track  at  rear  end  of  track. 
During  the  second  period,  the  'inertia  couples  become 


800 


f?5 ACT/QMS  ON  77PP/MG  PdfTTS 
DOUBLE  WML  SYSTfM 


-  fff/KT/OMS  ON  71PPM6  f>AffT3  W  B^TTfffY— 


-  ftf ACTIONS  ON  T/PP/MG  P/4ffT3  OVT  Of  BdTTtffY— 


Fig. 6 


801 

dvc  dvc 

reversed,  therefore,  A-D-E-Br—  sine.  xp-m_ — — 

dt  dt 

cose  y_-G-H-I-J-K=0  where  —7*-  is  determined  by 

dt 

the  relation, 

TDri       ,         dvc 
3  (irr-»-mc+me  ) 

where  mr=mass  of  recoiling  parts,  mc=mass  of 
caterpillar  and  mount  excluding  recoiling  parts, 
me=  equivalent  mass  for  rotational  inertia. 

ELEVATING  ARC  AND  TRUNNION  REACTION     In  comput- 
OF  TH£  TIPPING  PARTS.  ing  the  various 

reactions  in  a 
double  recoil 
system,  we 

oust  consider  the  inertia  effect  of  the  various 
parts  in  modifying  these  reactions  over  their 
static  values  or  as  would  occur  with  a  single  re- 
coil. 

The  primary  inertia  forces  induced  by  the 

double  recoil  are: 

For  the  upper  recoiling  parts: 


(1)  The  inertia  force  of  the  upper 
recoiling  mass  divided  into  components 
through  the  center  of  gravity  of  the 
upper  recoiling  parts,  parallel  and 
normal  to  the  axis  of  the  bore, 
respectively. 

For  the  lower  recoiling  mass: 

(2)  The  inertia  force  of  the  top 
carriage  and  cradle  acting  through 
the  center  of  gravity  of  this 
combined  mass,  opposite  to  the  ac- 
celeration, and  parallel  to  the  in- 
clined plane. 

The  inertia  resistance  of  the  top  carriage 
and  cradle  may  be  divided  into  two  parallel 

coapnents  through  the  center  of  gravity  of  the 
cradle  and  top  carriage  respectively  the 


802 


magnitude  of  the  components  being  proportional 
to  their  respective  masses. 

xt  and  yt*  coordinates  of  upper  recoiling 
parts  parallel  and  normal  to 
the  axis  of  bore. 

x  and  y  -  coordinates  of  lower  recoiling 
parts  parallel  and  normal  to 
inclined  plane  . 

Kx   =  inertia  component  along  bore  of  upper 
recoiling  mass  through  its  center 
of  gravity  (Ibs) 

Kv  =  inertia  component  normal  to  bore  of 
i 

tipper  recoiling  mass  through  its 

center  of  gravity  (Ibs) 

Kxc  =  inertia  force  of  cradle  through  its 
center  of  gravity  and  parallel  to 
inclined  plane  (Ibs) 

Kxc  =  inertia  force  of  top  carriage  through 
its  center  of  gravity  and  parallel 
to  inclined  plane  (Ibs) 

Wr  =  weight  of  upper  recoiling  parts  (Ibs) 
xr  and  yr  =  coordinates  from  trunnions  of 
center  of  gravity  of  upper 
recoiling  parts  in  battery, 
parallel  and  normal  to  bore  (ft) 
HC  =  weight  of  cradle  (Ibs) 

x-   and  y«  =  coordinates  from  trunnions  of 
t       i 

cradle  parallel  and  normal  to 

bore  (ft) 
Wc   =  weight  of  top  carriage  (Ibs) 

Hc  =  total  weight  of  lower  recoil  parts  (Ibs) 

W -»  ~"  W  A   TW 

1     * 

w"t=  weight  of  tipping  parts 

X^  and  Yt  =  components  of  trunnion  reactions 
parallel  and  normal  to  axis  of 

bore  of  gun. 
£  =  elevating  arc  reaction  (Ibs) 

• 


803 


j  -  elevating  arc  radius  about  trunnions 

or  perpendicular  distance  to  line  of 

act  on  of  £  (Ibs) 
B  =  total  braking  between  upper  and  lower 

recoiling  parts  (Ibs) 
RI  =  total  friction  between  upper  and 

lower  recoiling  parts  (Ibs) 
P  =  total  resistance  between  upper  and 

lower  recoiling  parts  (Ibs) 
N  =  total  normal  reaction  between  upper  and 

lower  recoiling  parts  (Ifas) 
Z  =  relative  displacement  of  upper  recoil- 

ing parts  wit})  respect  to  lower  recoiling 

parts. 

Pjj  =  powder  reaction  on  base  of  breech  (Ibs) 
e  =  distance  from  P  to  center  of  gravity  of 

upper  recoiling  parts  (in) 

Then  during  the  acceleration  for  the  upper  re- 
coiling parts,  we  have 

d«x 
Ph  »(B+R  -W_sin0)=mr  -  -  along  the  bore  and  the 

dt* 

external  reaction  on  the  upper  recoiling  parts 
during  the  powder  acceleration,  becomes 

d»xt 
Kx  =Pb-mr    g  (Ibs)  along  the  bore 

1.  U  L 

=B+R-Wrsin0  along    the  bore 

=P-Wrsin0  along    the  bore 

During    the   retardation, 


Br  -  =  -(B+R-tfrsin0)  and  the  external  reaction 

dt2  on  the  recoiling  parts 

parallel  to  the  bore  is  the  inertia  force, 

d»xt 

Ktf  -  -m-  -  =B+R-tfPsin0 
x*     P  dt« 

=  p-wrsin0 
Hence  either  during  the  acceleration  or  retardation 


804 


the  external  com.ponent  parallel  to  the  bore  on  the 
recoiling  parts  equals  the  total  resistance  to 
recoil  off  the  upper  recoiling  parts..   The  in- 
ertia force  normal  to  the  bore,  becomes, 


t  t 

Kv  3mr~TT7   (Its)    Since  -  —  =  v  sin(0+9),  where 

*  t    at*  at 

v  is  the  velocity  of  the  lower  recoiling  parts 
up  the  plane  . 

Kv  =  mr  &  sin(0+6)  (ibs) 
y»    r  dt 

=  N-Wrcosd        (Ibs) 

For  the  lower  recoiling  parts,  we  have 

dv 

K«s=  -m  —  (Ibs)  along  the  inclined  plane 
»     *dt 

Kxc  '-rag  -->  (Ibs)  along  the  inclined  plane 

a     2  u  t 

Elevating  arc  and  trunnion  reactions: 


Let  us  now  consider  the  tipping  parts,  that 
is  the  recoiling  parts,  together  with  the  cradle. 

By  the  use  of  D'Alemberts  principle  the 
problem  in  Kinetics  is  reduced  to  one  of  statics, 
provided,  we  introduce  the  proper  inertia  forces.. 

Further,  the  nutual  reaction  between  the 
upper  recoiling  parts  and  the  cradle  of  the 
lower  recoiling  parts:,  becomes,  an  internal  force 
for  the  system  consisting  of  the  tipping  parts. 

Therefore,  introducing  the  inertia  forces, 
we  have: 

For  the  reactions  of  the  tipping 

parts  in  battery:- 
Along  the  bore:  fig.  (6) 


t 

+Ecose9+Wtsin0-Kxcicos(0+9)-2X=0 
dt8 


805 


Normal  to  the  bore: 

Wtcos0-E  sine-Kxcisin(9+ef)+Kyi-2Y»0 

Moments  about  the  trunnion: 
d«xt 


(0+9)xc  »-Ej  -  0  since  in  the  battery  position 
the  center  of  gravity  of  the  tipping  parts  is 
located  at  the  axis  of  the  trunnions.   Further, 


t 

pu-n-  -  =B+R-WPsin0*K,  i   We  have,  for  the 
dt2 

elevating  arc  reaction, 

Pbe+Kxiyr+Kvixr+Kxci  [ycicos(0+6)-xc  is  in  (0+6)] 
E=  -  '  - 

-j 

and  for  the  components  of  the  trunnion  reaction 
2X=Kxi+Scos9e+Wtsin0-Kxcicos(0+8) 

2X»Wtcos0+Kyi-E  sinee-Kxeisin(0+6) 

For  the  reactions  of  the  tipping  parts  out  of 
battery: 

In  any  intermediate  position,  out  of 
battery  the  entire  tipping  parts  are  displaced 
backwards  up  the  inclined  plane  but  in  addition 
we  have  a  relative  displacement  between  the  re- 
coiling parts  and  the  cradle  of  the  top  carriage, 
equal  to  Z  (in). 

Therefore,  the  moment  of  the  tipping  parts 
about  the  trunnions,  become  Wr(lr+Zcos0)+Hcilci=Mt 

where  lr  and  lc  i  are  the  horizontal  coordinates 

of  the  upper  recoiling  parts  and  cradle  in  the 
battery  position.   Since  center  of  gravity  of 
the  tipping  parts  are  located  at  the  trunnion  in 
the  battery  position,  we  have  V?rlr+Wcilc  i=0 
hence  Mt=WrZcos0.   Then,  the  reactions  along  the 
bore 


806 

Kxi+Ecos9e+Wtsin0-Kxcicos(0+9)-2X=0 
Horaal  to  the  bore: 

Wtcos0+Ky i-Esin6e-Kxc i sin (0+9 )-2Y=0 
Moments  about  tbe  trunnion: 

K,iyr-»Ky  iXr+Kxcicos(9+0)yc,-Kxc.sin(9+0)xc,+Wrxr 

cos0-  Ej=0 

Hence,  we  have  for  tbe  elevating  arc  reaction  for 
a  relative  displacement  Z  out  of  battery 


j 
and  for  the  components  of  tbe  trunnion  reactions, 

2X=KX i+Ecos6e+Wtsin0-Kxcicos(6+0) 
2Y=Htcos0+Kyi-Esin6e-Kxcisin(6+0) 

REACTION  BETWEEN  UPPER  AND     In  the   calculation 
LOWER  RECOILING  PARTS.      of  guide  and  clip  re- 
actions and  the  bend- 
ing stresses  in  the 

cradle  it  is  necessary 

to  know  the  nature  of  the  reaction  between  tbe 
upper  and  lower  recoiling  parts  as  well  as  its 
distribution. 

The  reaction  between  tbe  two  recoiling 
masses,  consists  of: 

(1)  Tbe  resultant  braking  reaction 
acting  parallel  t o  the  guides 
and  through  the  controid  of  the 
various  pulls. 

(2)  The  guide  friction  acting  along 
the  guides. 

(3)  The  normal  clip  reactions,  which 
may  be  divided  into: 

(a)     a  normal  component  per- 
pendicular to  the  axis  of  tbe 


807 


our  or  B/trrf/?y 


Fig.7 


808 


ft&ICTMN  B£TW££W  UPPfft  4N£>  tOWfff  fffCOtUMG  FMffrS  : 
POUBL  E  fffCOfi 


F>OS/r/OM 


Fig.  5 


809 


bore, 
(b)     a  couple  between  the 

two  parts. 

The  magnitude  of  the  couple  depends  upon  the 
assumed  position  of  the  line  of  action  of  the 
normal  component;  therefore,  we  may  assume  the 
normal  component  in  its  most  convenient  position 
for  calculation.   Let 

N  *  total  normal  reaction  between  upper  and 

lower  recoiling  parts  (Ibs) 
Nt  *  front  normal  clip  reaction  (Ibs) 
Nf  *  rear  normal  clip  reaction  (Ibs) 
xt  and  yt  »  coordinates  of  front  clip  re- 
action along  and  normal  to 
bore  with  respect  to  center  of 
gravity  of  upper  recoiling  parts 
(in) 
xc  and  ya  *  coordinates  of  rear  clip  reaction 

(in) 
M  »  couple  or  moment  reaction  between  upper 

and  lower  recoiling  parts  (inch-  Ibs) 
Pn*total  hydraulic  pull  including  packing 

friction  (Ibs) 
Pa*  total  recuperator  reaction  including 

packing  friction  (Ibs) 
R  *  total  guide  friction  (Ibs) 
dbs  distance  from  center  of  gravity  of 

upper  recoiling  parts  to  PD  (in) 
da*  distance  from  center  of  gravity  of  upper 

recoiling  parts  to  Ba  (in) 
dr  »  distance  from  center  of  gravity  of 

upper  recoiling  parts  to  R 
n  «  coefficient  of  guide  friction  (0.15  ap- 

prox. ) 

B  *  2Pn+2Pa»Total  braking  (Ibs) 
R  =  n(Nt+Nf)=guids  friction  (Ibs) 
lv  *  horizontal  distance  from  rear  roller 

contact  of  top  carriage  and  inclined 


810 


plane  to  line  of  action  of  Wr 
d  =  distance  from  A,  normal  to  lino  through 

center  of  gravity  of  upper  recoiling 

parts  and  parallel  to  bore 

Then  B  db»ZPhdh+ZPada.   Considering  the  re- 
actions on  tbe  recoiling  mass  in  battery,  we 
have,     dtx 

Pb-m,   *  =B+R-Wrsin0,  along  tbe  bore 
dt 

N=Ky  i+Wrcos#,  normal  to  tbe  bore 
M=Pbe+Bdb+R  dr,  moments  about  center  of 

gravity. 

Taking  moments  about  A  fig.  (7)    at  the  rear 
roller  contact  of  top  carriage  and  inclined 
plane,  we  have,  for  tbe  moment  of  the  re- 
action exerted  by  the  upper  recoiling  parts, 
on  the  lower  1 

M  =B(d-dh)+R(d-d,)+M-N(  —  ^  +dtan0) 

costf 

Substituting  for  M,  its  value  M=Bdb+Rdr+Pbe 
and  for  N,  its  value  N=Wrcos0<Kyi  we  have 

Ma=Pbe+(B+R-Wrsin(?-Kyitan0)d 
Ki 


r        r 

COS0 

Hence,  the  reaction  of  the  upper  recoiling  part 

on  the  lower,  during  tbe  powder  period,  is  equivalent 

to: 

(1)  A  couple  Pbe 

(2)  A  force  through  the  center  of 
gravity  of  the  recoiling  parts 
parallel  to  the  bore:  B+R-Wrsin0-Ky  itan0 

(3)  A  vertical  force  through  the  center 
of  gravity  of  the  recoiling  parts, 

Kv. 

Wr+  —  — 
cos0 

After  the  powder  period,  the  reactions  on  the 
recoiling  parts,  become 


811 


-m   ,J.,~B+R-W,sia(y,  along  the  bore 
r  dt 

N-Kyi+Wrcos0,  normal  to  the  bore 
M=Bd^+Rdr, moments  about  the  center  of  gravity 
Taking  moments  about  A,  fig. (8)  at  tbe  rear 
roller  contact  of  top  carriage  and  inclined 
plane,  we  have,  for  tbe  moment  of  the  reaction 
exerted  by  the  upper  recoiling  parts,  on  tbe 
lower,  the  recoiling  parts  having  a  relative 
displacement  Z.        1  -Zcostf 

M*=B(d-dh)+R(d-dr)+M-N( —  +  d  tan0) 

cos0 

Substituting,  as  before,  for  M  and  N,  we  have 

V 

MA=(B+R-WPsin0-Kuitan0)d-(WP+-i— -)(l-Zcos0) 
y         r  cos0 

Hence,  tbe  reaction  of  the  upper  recoiling 
parts  on  the  lower,  after  the  powder  period, 
that  is  during  the  retardation  of  tbe  upper  recoil- 
ing parts,  is  equivalent  to: 

(1)  A  force  through  the  center  of 
gravity  of  the  recoiling  parts 
parallel  to  the  bore 

B+R-Wrsia0-Ky i tan0 

(2)  A  vertical  force  through  the 
center  of  gravity  of  the  recoil- 
ing parts: 

.,.  ,„,„'  Br+  V_ 

cos0 

Tbe  mutual  reaction  between  tbe  upper  and  loner 
recoiling  parts,  can  be  determined  immediately  as 
follows: 

(1)  B+R  along  the  bore 

(2)  N    normal  to  the  bore 

(3)  M    a  couple  between  the  parts 
Now,  N=Wrcos0+Ky i (algebraically) 


K; 


fr-Wrsin0+ — Kyitan0   (vectorially) 

CQSJ0 


812 


and  through  the  center  of  gravity  of  the  recoiling 
parts.   Further  B  and  R  may  be  resolved  into  a 
vector 

B+R  parallel  to  B  and  R  through  the  center  of 

gravity  of  the  recoiling  parts  and  a  couple 
Bdb+Rdr.  Hence  combining  B+R  and  M,  we  have 

B+R+Bdb+Rdr+M=B+R  (through  the  center  of  gravity 

of  the  recoiling  parts,  parallel 

_  to  the  bore) 

since  M=-Bdb-Rdr 

Combining,  the  parallel  components  through  the 
center  of  gravity  of  the  recoiling  parts,  we  have 
B+R-WrsingJ-Kyitan0,  along  the  bore 

Kyi 

Wr+  vertically 

COS0 

-(bdb+Rdr)         a  couple  between  the  parts 

Guide  and  clip  reactions: 

For  the  front  clip  reaction,  we  have 

v»«-- 1 

Bdb+Rdr 
=(Wrcos0+Ky i)»t~(       )  (Ibs)  acting  upward 

on  recoiling 

parts,  and 

for  the  rear  clip  reaction,  N  *Nx  *- 

Bdb+Rdr 

»(Wrcos0+Ky i)xt+( — ')  (Ibs)  acting  up- 
ward on  recoil' 
ing  parts. 

VARIABLE  BRAKING  ON  UPPER     Usually  we  have  a 
RECOIL  BRAKE  DOUBLE  RSCOIL  given  upper  recoil 
SYSTEM.  system  with  a  constant 

braking  for  the  lower 
recoil  system,  since 

given  single  recoil  mounts  are  converted  into  a 
double  recoil  system  by  allowing  the  top  carriage 
to  slide  along  an  inclined  plane.   Further,  in  the 


813 


design  of  a  double  recoil  system,  since  at  high 
elevations  of  the  gun,  the  component  of  the  re- 
action of  the  upper  recoiling  parts  along  the 
plane  is  small,  the  movement  up  the  plane  there- 
fore, becomes  relatively  small.   Hence  in  a  design 
layout  the  throttling  grooves  of  the  upper  recoil 
system  nay  be  calculated  on  the  basis  of  a  single 
recoil  at  maximum  elevation. 

We  have  therefore  a  very  important  class  of 
double  recoil  systems,  where  the  upper  recoil 
throttling  is  based  on  a  single  or  static  recoil 
and  the  lower  recoil  braking  is  designed  for  an 
approximate  constant  resistance  at  minimum  ele- 
vation. 

The  recoil  braking  of  the  upper  recoil  system 
consists  of  the  sum  of  the  following  components: 

(1)  The  recuperator  reaction,  which 
is  a  function  of  the  relative  dis- 
placement between  the  gun  and  top 

carriage. 

(2)  The  throttling  reaction,  which 
is  proportional  to  the  square  of 
the  relative  velocity  at  a  given 

relative  displacement. 

(3)  The  guide  and  packing  frictions, 
which  depend  upon  the  normal  reaction 
between  the  parts,  etc.,  but  can  be 
assumed  approximately  constant. 

The  lower  recoil  braking  will  be  assumed 

constant  at  minimum  elevation. 

Considering  fig.  (9) 
Let 

Pb*  powder  reaction  (Ibs) 

B  *  total  braking  (Ibs) 

Pf  *  total  friction  (assumed  constant)  (Ibs) 

Ph  «  hydraulic  braking  (Ibs) 

p  »  Pa+Ph+Pf 


814 


Png  =  hydraulic  pull  fron  static  force  diagram 

Pa=  recuperator  reaction  (Ibs) 

N^normal  reaction  between  upper  and  lower  recoil- 
ing parts  (Ibs) 

R  »  brake  reaction  of  lower  recoil  system  (Ibs) 
0=  angle  of  elevation 

9*  angle  of  inclination  of  inclined  plane 
Wr=  weight  of  upper  recoiling  parts  (Ibs) 
n*c=  weight  of  lower  recoiling  parts  (Ibs) 
Vf=  free  velocity  of  recoil  (ft/sec) 
Vr»absolute  velocity  of  recoiling  parts  parallel 

to  axis  of  gun  (ft/sec) 
Vc=  velocity  of  lower  recoiling  parts  along  the 

inclined  plane  (ft/sec) 
Z  =  relative  displacement  between  upper  and  lower 

recoiling  parts  (ft) 
Xc  =  displacement  of  top  carriage  up  inclined  plane 

(ft) 

Xr  =  absoluts  displacement  of  gun  parallel  to  axis 
of  bore  (ft) 

B'=  counter  recoil  buffer  rsactnon 

During  the  powder  pressure  period: 


On   the   upper   recoiling   parts,    we 
-p+Wrsin0=BrLJLL       (l)along   the   bor 


Pb 

dt« 


N-Wrcos0=mP (2)   normal   to   the   bore. 

dta 

On  the  lower  recoiling  parts,  we  have 

daxc 

Pcos(0+e)-Nsin(0+e)-R-Hcsine»nic (3)  up  the 

dt8 

inclined  plans.   We  have  further,  the  following 
kinematical  relations:- 

dxr 

«*^^«"«^E  v  +  w/*rtof0?  +  fti  Y~-Y  A  Y 

V 1* —  i       ^^  Vy»o"lTV^COS\iL/Toy  Aw»"™Ay»ol*A/» 

'I      ^j-^  tCi         u  r         i  c -L         C 

— -  =   Vcsin(0+e)  :          Yr=ycsin(0+6) 


815 


d*xp  dvrel   dv 
hence  -  =  -  +  —  cos  (0+6) 
dt2    dt    dt 

d*yr  dvc 

-  =  -  sin(0+6) 
dta  dt 

Now  between  any  two  instants  tn_j  and  tn  we  have, 
from  eq.  (1) 

tn  Pbdt    p-Hrsin0 

—  ~  (—  ^  -  )(tn-tn-l>'V?  -V?-1 
mr       mr 

. 
p-W 


r  p-W_sinCf 

therefore  Vj-Vj'1  +(Vfn-Vf  ^^-(-g-  -  )A  tn 

which  gives  a  "point  by  point"  method  for  determin 
ing  the  absolute  velocity  of  the  gun  parallel  to 
the  bore. 

Now  if  we  substitute  for  the  normal  reaction 
N  in  (3)  its  value 

dvc 
N  *t»rcos0+inr  —-  sin(0+6) 

u  t 

we  have 

P  cos(0+9)-Wrsin(0+6)cos0-Br 

dt 


dt 
hence 

dvc   p  cos(0+6)-Wcsine-Wrsin(0+6)cos0-F 
dt       m  +m_sin*(0+9) 

w    i 

and  between  instants  tn-i  and  tn,  we  have 


mc+mrsin«(0*9) 


A  t 


The  total  braking  P,  becomes  P^P^Pw  +  Pf   (Ibs) 
In  the  static  or  single  recoil,  the  top  car- 


816 


riage  stationary,  we  have  Pns»Co  ""* — 

"xn« 
NOII  for  the  same  relative  displacement  between 

the  upper  and  lower  recoiling  parts  for  the  double 
recoil,  the  throttling  area  is  the  same,  namely 

wxn,  then       _«  v 

vrel  vrel* 

Ph=co  —   bence  Ph=Phs  — — 
"In  *• 

Therefore,  from  a  static  force  diagram, 
knowing  the  relative  displacement  =  static  recoil 
displacement,  we  may  determine  Phs  and  v| .   If 
'1TS1  has  been  determined  for  the  point,  the 
hydraulic  braking  is  readily  determined  from 
the  above  equation. 

The  recuperator  reaction  is  determined  from 
the  static  force  diagram  when  the  relative  dis- 
placement is  known.   When  the  upper  recoiling 
parts  begin  to  counter  recoil  relatively  to  ths 
lower  recoiling  parts,  we  have 


v«   "Pf 

Procedure  for  recoil  calculations 

We  must  first  construct  a  static  force  and 
velocity  diagram  for  the  upper  recoil  system  as 
would  occur  if  the  mount  had  a  single  recoil, 
the  top  carriage  being  fixed.   Let 

v0  *  nuzzle  velocity  (ft/sec.) 

u*  travel  up  bore  (ft) 

w  =  weight  of  projectile  (Ibs) 

w  =  weight  of  charge  (Ibs) 

Pa=max.  total  powder  reaction  (lbs)wv» 

then  average  pressure  on  breech  Pe  *   — ~~*  (Ibs) 

2gu 

Pressure  on  breech  when  shot  leaves  muzzle  — 

27       u 

Pob  «  —  b«  1.18  PB   (Ibs) 

4     (b+u)» 


.here  b-(2Z  £  -  l)t  /I-  g  £>•  -1   (ft)   ^  „ 

3 

Time  of  travel  to  bore  to  3  T~  uo     (sec) 

wv0+4700w 
Max.  free  velocity  of  recoil  Vf  *  -  (ft/sec) 

"r 
Free  velocity  of  recoil  when  shot  leaves  muzzle  — 

(w+0.5w)v0 

(ft/sec) 


Time  during  expansion  of  powder  gases  — 
2(Vf-V0)   wr 

ti°=  —  r  (sec) 

Total  powder  period  T  =  t.  +t0  (sec) 

*o 

Free  displacement  of  recoil  during  travel  up  bore  — 

w+0.5w 


Free  displacement  during  expansion  of  gases  •— 

Pob    (T-tQ)« 

«f -o  -  —  «  3 +vfo<T-t0 

r 

Total  displacement  of  free  recoil  during  powder 
period —  B=xfo*xf'o 

Three  points  are  sufficient  to  establish  approximate 
the  velocity  curve  during  the  powder  period.  They 
•ay  be  taken  at  times  to  t\m   and  T  respectively. 

The  total  resistance  to  recoil  for  constant 
resistance  to  recoil, 

;•>*! 


b-E+VfT 

for  variable   resistance   to  recoil 
mrV|+m(b-E)« 

2[b-E+VfT-  |   —  (b-E)] 


818 


At  t0  when  the  shot  leaves  the  nuzzle  — 

.  Mo 

Vo~Vfo~  mr 


At   t,jj  when  ire   have   max.    restrained   recoil 
velocity,  Koto 

vnTvfn  --   (ft/sec) 
•r 

K  ta 


where  Vfm»Vf  Q+Pob  (tm-tQ)tl  --  ]  (ft/sec) 

4fflr(Vf-V0) 


K(T-t0) 
tm  «  T (sec) 

pob 
At  tine  T,  the  end  of  the  powder  period  — * 

K  T 
Vr=Vf-  —  (  ft/sec) 

mr 

K0T2 

Er=X£ (ft) 

2mr 

After  the  powder  period,  during  the  retardation, 
we  have  for  constant  resistance  to  recoil, 


V  =  /— £(b-X)    (ft/sec) 
mr 

for  variable  resistance  to  recoil, 


/(K  -  f(b+X-2Er)(b-X)) 

v  =  / 2_2 £ (f 

mr 

where  b  =  the  total  length  of  static  recoil  (ft) 


819 


a  =  Cs  —  ;   Cs=0.85  approx.;   b=  perpendicular 

distance  froa 

spade  to  line  of  action  of  K.   --* 
Construction  of  static  force  diagram: 

We  have,  for  a  constant  resistance  through- 
out recoil,  K*Pbs+Pa+Pf-Wrsin0  hence  Phs*pa*pf= 
K+Wrsin0  (a  constant) 
For  variable  recoil,  in  battery  K  SKO,  out  of 

batteryK  =k 
where  k  =  Ko-m(b-Er)  and  K  =  KQ  during  the  powder 

period  . 

=K0-m(x-Er)=k+m(b-x) 
hence  Pbs+Pa+Pf  =K0+tfrsin0-fl>(X-Er) 

r 

r 

Value  of  components  Pf,Pa  and  P0.   For  a  first 
approximation,  the  friction  component  becomes, 
Pf=0.2Wrcos0+p  (estimated  packing  friction)  and 
will  be  assumed  constant.  The  recuperator  re— 
action  becomes,    p  _p  . 

Pa=Pai+  -  X  for  springs 
a   <*±      jj 

where  Pai=total  initial  spring  reaction 
Pa£=total  final  spring  reaction 

Vo   k 
Pa=pai  (  -  )   "here  k=l.l  to  1.3 

v0-v 

Vrt=initial  volume  (cu.ft) 


rt 
0 


• 


i  i 

8 


Av  =  effective  area  of  recuperator  piston 

(sq.in) 

Pai=  initial  air  pressure  (Ibs/sq.in) 
mo  =  ratio  of  compression  (from  1.5  to  2) 
The  hydraulic  throttling  reaction,  becomes  for 
constant  recoil,  Pns=(K+Wrsin0)-Pa-Pf 


820 


for  variable  recoil  Pns=K0+Wrsin0-m(X-Er)-Pa-Pf 

where  the  value  of  Pa  corresponds  to  the  displace- 
ment X. 
Construction  of  static  counter  recoil  diagram: 

The  counter  recoil  may  be  divided  into 
and  acceleration  period,  controlled  or  regulated 
by  a  throttling  resistance  through  a  constant 
orifice,  and  the  retardation  period  where  the 

recoiling  mass  is  brought  to  rest  into  battery 

by  a  constant  resistance  to  recoil,  with  a  varying 

buffer  throttling.   If 

Pa=tbe  recuperator  reaction 

Pf-total  friction  of  counter  recoil  assumed 
the  same  as  for  recoil  and  constant. 

Bs=static  buffer  reaction 

lo=length  of  constant  orifice  period  (ft) 

lb=length  of  variable  orifice  period  (ft) 
Then  during  the  acceleration, 

dv  vs 

pa-Pf-Wrsin0-B^=mrv  —  where  B^=  co  —  («0=  a  con- 

U  X  «M£ 

0  stant) 
and  during  the  retardation 

_  f*   I  TJ  X 

dv          covs 
B£  +Wrsin  0  +  Pj  -Pa  =  -  mr  v  —  where  B^=  — — 

•i  "* 

Now  —  nay  be  determined  by   assuming    a  max. 

wo 

counter  recoil  velocity  »  3.5  ft/sec. 

at  max.  velocity,  we  have, 

e  i 

Pa-Pf-Wrsin0-  (-r)vjs  =  0   and  assuming  v*s, 
»5  c, 

we  readily  determine   —  *  G 

* 

The  velocity  and  force  curve  during  the  first 
period  may  therefore  be  constructed  as  follows: 
(1)     Plot  the  recuperator  reaction 
against  counter  recoil  displace- 
ment, that  is, 


821 


V*, 


V0-Av(b-X) 

b  =  length  of  re- 
coil 

(2)  Assume  P£  =(0.2Wrcos0) 

(estimated 

packing  friction)  Con- 
stant for  the  counter 
recoil. 

(3)  Divide  the  acceleration  period 
into  "n"  intervals  and  take  the  mean 
air  pressure  for  this  interval. 
Then,  knowing  the  velocity  at  the 

beginning  of  the  inta1"**!*  *e  can 
compute  the  velocity  at  the  end  of 
the  interval  by  the  formula,  - 


log  (A-  -— )slog(A-  -j-^) p 

where  A  =  Pa-Pf-Wrsin0 

co 

—  =  G  and  determined  as  outline  above. 


(4)     .Next  construct  from  the  velocity 
curve  a  static  buffer  against 
counter  recoil  displacement,  that 
is 

co 

Bs'(— )  v« 
»o 

The  velocity  and  force  curve  during  the  re- 
tardation period  of  counter  recoil  may  be  con- 
structed, as  follower- 
CD     The  total  resistance  to  counter 
recoil  being  assumed  constant 
during  this  period,  we  have 
Bs+Wrsin0+Pf-Pa=Kv  whence,     /  2Kv(b-x) 


v  =  /  — 


822 


A/ 


rig. 9 


823 

-  m  v* 

T*  m 

where  Ky  =  and  vm  is  determined  from 

1^     the  previous  point  by  point 

method  to  the  end  of  the  dis- 
placement 10.   Then,  the  velocity  and  buffer  force 
against  recoil  displacement  is  determined,  since 

Jrarvm             vo    k 
B^  =  J_£JL_  +p  ( 2 ]K  -Pf-Wrsin0 

V0-Ay(b-X) 


/  2Kv(b-X) 
and  v«  / where  K 


v 


Dynamical  equations  of  double  recoil  for  point  by 
point  method  of  procedure  for  construction  of  re- 
action and  velocity  plots: 


Let  0  =  min.  angle  of  elevation  of  gun 

P  =  total  pull  between  upward  and  lower  re- 
coiling parts  (Ibs) 
Phg-  static  hydraulic  pull  (Ibs) 
Pa=  recuperator  reaction  (Ibs) 
F£=  total  friction  assumed  constant  (Ibs) 
Vj=free  velocity  of  recoil (ft/sec) 
Vr=velocity  of  upper  recoiling  parts  parallel 

to  upper  guides  (ft/sec) 
Vrej=  relative  velocity  between  upper  and 

lower  recoiling  parts  (ft/sec) 
V*o=  velocity  of  lower  recoiling  parts  up 

plane  (ft/sec) 
X  =  displacement  of  top  carriage  up  inclined 

plane  (ft) 
B£  =  static  counter  recoil  buffer  reaction 

(Ibs) 

R  =  lower  recoil  reaction  parallel  to  in- 
clined plane  (Ibs; 
Then,  during  the  powder  pressure  period, 

'+Pa+Pf          (1) 


824 


P-W  sin0 
Vg  =•  YJT1   (V£  -  Vg"1  ) • At    (2) 

mr 

[Pcos(0+8)-Wcsin9-R-Wrcos0sin (0+8)j At 

vn  =  ya-1  + (3) 

mc+mrsina (0+9) 

Vrel=Vr-Vccos(CJ+e)  (4) 

•-  — — — ^-  A  t  (5) 


+  -£ A  t  (6) 


c  "c 

2 
After  the  powder  period, 

P-WPsin0 
vn  „  vn-i  _  £ A  t 


After  gun  begins  relative  counter  recoil, 


p=p- 


a-     )_  P  ( 

V8 

In  determining  PQSvs  and  Pa  the  relative 

displacement  must  be  equal  to  the  static  dis- 

placement of  the  recoil,  that  is  *rel=xs»  from 
which  we  determine  P   v 


825 


240     M/M     HOWITZER,     GAS-ELECTRIC     T7PB,      DOUBLE     RECOIL, 


24°    Elevation,     R    •     45,000    Ibs. 

VELOCITY 

DISPLACEMENT 

P 

I 

T 

T 

6 

c 

R 

R 

u 

0 

n 

0 

0 

u 

a 

e 

e 

p 

i 

t 

t 

t 

n 

r 

1 

1 

n 

e 

a 

a 

r 

A 

a 

t 

r 

1 

1 

P 

i 

t 

t 

p 

a 

V 

• 

a 

I 

i 

1 

* 

T 

B 

r 

g 

V 

V 

a 

1 

i 

r 

a 
1 

e 

e 

e 

n 

" 

e 

a 
k 

1 

7° 

e 

a 
e 
o 

8 

i 

n 
g 

e 

i 

p 

f 
t. 

0 

o 

i 

o 

F 

a 

n 

o 

6 

n 

d 

r 

u 

e 

• 

c 

n 

e 

8 

1 

i 

d 

a 

During    Powder    Pressure    Period 

I 

.  OO4 

151700 

15.93 

.703 

15-332 

.0307 

.0014 

2 

.006 

142400 

32.849 

1.  648 

31.438 

.  1710 

.0084 

3 

.01 

140000 

41.431 

3.310 

38.  593 

.5217 

.0332 

4 

.012 

138300 

41.  351 

5.  105 

36.971 

.9751 

.0837 

5 

.016 

128900 

37.351 

7.225 

31.  161 

1.  5201 

.  1824 

6 

.02 

117900 

32.  801 

9.405 

25.051 

2.  082 

.  3487 

7 

.02 

108000 

28.  65 

11.195 

19.061 

2.523 

.5548 

8 

.02 

96000 

24.99 

12.  51 

14.  271 

2.  856 

.7919 

9 

.02 

86500 

21.  721 

13.45 

10.  191 

3.  101 

1.051 

10 

.02 

77ioo 

18.835 

14.  03 

6.  8O5 

3.271 

1.  326 

11 

.02 

677oo 

16.33 

14.  23 

4.  12 

3.3804 

1.  6091 

12 

.02 

61000 

14.  10 

14.175 

1.95 

3.  441 

1.893 

13 

.02 

56600 

12.052 

13.941 

.092 

3.  461 

2.  174 

14 

.002 

54900 

11.  854 

13.911 

.076 

3.  46 

2.  202 

Gun    beginning    to     C'Roooil 

15 

.01 

46300 

11.03 

13.  601 

.64 

3.457 

2.339 

16 

.01 

45430 

10.233 

13.  263 

l.  137 

3.449 

2.474 

17 

.01 

42300 

9.  501 

12.  86 

1.  519 

3.436 

2.  605 

18 

.01 

38790 

8.841 

12.  397 

1.  779 

3.419 

2.731 

19 

.01 

35680 

8.  243 

11.  868 

1.937 

3.  40 

2.852 

20 

.0  1 

33550 

7.  689 

11.  297 

1.  981 

3.38 

2.  968 

21 

.  01 

33070 

7.  144 

10.  717 

2.056 

3.36 

3.078 

22 

.01 

31600 

6.63 

lo.  10  7 

2.03 

3.34 

3.  182 

826 


24O  M/M  HOKITZER, GAS-ELECTRIC  TYPS, DOUBLE  RECOIL, 


'  o  o  r.  t  i  nu 

ed) 

23 

.01 

31*740 

6.  113 

9.5 

2.037 

3.32 

3.  28 

24 

.01 

31500 

5.  6oi 

8.  889 

2.0  19 

3.3 

3.372 

25 

.0  U 

3168O 

4.  982 

8.  159 

2.018 

3.276 

3.474 

26 

.014 

31350 

4.  269 

7.  299 

1.981 

3.  248 

3.582 

27 

.016 

31590 

3.  446 

6.323 

1.974 

3.  216 

3.  690 

28 

.  0  18 

31360 

2.530 

5-215 

1.94 

3.  181 

3.794 

29 

.02 

31490 

1.  §O6 

3.99 

1.914 

3.  142 

3.886 

30 

.02 

31520 

.  481 

2.765 

1.889 

3.  104 

3.954 

31 

.01 

31470 

.031 

2.  153 

1.878 

3.085 

3.979 

32 

.02 

31350 

1.O5 

1.063 

1.962 

3.047 

4.011 

33 

.016 

30000 

1.82 

.  036 

1.85 

3.017 

4.  O2 

240  M/M  H01ITZER,  TRACTOR  MOUNT,   DOUBLE  RECOIL. 


)°  Elevation,  R  -  80OOO 

VELOCITY 

DISPLACEMENT 

P 

I 

T 

T 

G 

C 

R 

U 

0 

i 

o 

o 

u 

* 

e 

P 

i 

t 

t 

t 

n 

r 

1 

n 

e 

a 

a 

r 

a 

t 

r 

1 

1 

i 

t 

P 

s 

V 

a 

i 

1 

a 

r 

B 

g 

V 

a 

1 

i 

r 

e 

e 

n 

• 

g 

• 

e 

e 

k 

6 

8 

i 

P 

e 

1 

n 

P 

t 

c 

e 

i. 

l 

5 

a 

r 

n 

o 

e 

r 

o 

e 

During  Powder  Pressure  Period 

1 

.004 

152000 

15.  76o 

.753 

15.011 

.O300 

.0015 

2 

.  O04 

144600 

28.  578 

1.  425 

27.  16O 

.  1143 

.0059 

3 

.  004 

143900 

35.505 

2.087 

33.428 

.2355 

.0129 

4 

.  004 

14O30O 

39.258 

2.713 

36.558 

.3755 

.0225 

5 

.004 

137600 

41.  235 

3.3C6 

37.  943 

.  5245 

.0345 

6 

.  OO4 

13460O 

42.035 

3.869 

38.  185 

.6763 

.0439 

7 

.003 

133500 

41.  355 

4.  984 

36.  400 

.9751 

.0834 

After  Powder  Pressure  Period 

8 

.02 

125800 

36.  225 

7.303 

28.965 

1.  6288 

.2072 

827 


240     M/M     HOWITZER, TRACTOR     MOUNT,     DOUBLE     EECCIL. 


(  Cent  inaed) 

9 

.02 

110800 

31.  "705 

8.803 

22.945 

2.  1479 

.3683 

10 

.02 

10050O 

27.605 

9.753 

17.905 

2.5565 

•  5539 

11 

.04 

91300 

20.145 

1O  .  6  2  1 

9.575 

2.  1063 

.9613 

12 

.02 

74400 

17.105 

10.138 

7.005 

3.  2722 

1.  1689 

13 

.02 

69000 

14.290 

9.360 

4.  980 

3-3920 

1.3640 

14 

.02 

64600 

11.  650 

8.340 

3.360 

3.4750 

1.  5410 

15 

.02 

61500 

9.  140 

7.150 

2.  040 

5.529 

1.  696 

16 

.02 

58300 

6.760 

5.790 

1.010 

3.560 

1.825 

ft 

.023 

56700 

4.  100 

4.  120 

.0 

3.5W 

1.939 

Gun    beginning    to     C'Eecoil 

18 

.004 

477oo 

3.710 

s.^so 

.  0 

.  1592 

1.955 

19 

.004 

477oo 

3.  320 

3.340 

.  0 

.1592 

1.969 

20 

.004 

477oo 

2.930 

2.950 

..O05 

.  1592 

1.982 

21 

.004 

477oo 

2.  §40 

2.  560 

-.010 

.1592 

1.993 

22 

.  004 

477oo 

2.  IgO 

2.  170 

-.010 

.  1592 

2.003 

23 

.  004 

477oo 

1.760 

1.780 

-.010 

.1592 

2.010 

24 

.004 

477oo 

1.370 

1.390 

-.010 

.  1592 

2.016 

25 

.  004 

477oo 

.980 

1.00 

.015 

.1592 

2.  021 

26 

.  004 

477oo 

.590 

.610 

.016 

.1593 

2.021 

2-7 

.  004 

477oo 

.  200 

.  220 

.019 

.1592 

2.026 

OP 
Plane 
(in) 

Kxt 

Kyt 

Kx 

Vs 

Fh<-£HH 

<1 

0 

145000 

45000 

62COO 

1180OO 

2 

124000 

32000 

46OOO 

96000 

4 

112000 

26000 

38000 

82000 

6 

10  3  5  CO 

21000 

30000 

7000O 

8 

9500O 

17OOO 

24000 

58000 

10 

89000 

13000 

18000 

49000 

12 

82000 

1OOOO 

14000 

400OO 

14 

76000 

7200 

11000 

32000 

16 

7OOOO 

5000 

77oo 

240OO 

18 

64200 

2100 

4  COO 

13000 

20 

60000 

1000 

1000 

130OO 

22 

55OOO 

-     2000 

-   1500 

8OOO 

24 

§2000 

-   3000 

-   2500 

5000 

26 

50000 

-     6000 

-    8500 

2000 

828 


Gun  beginning  to  Counter  Recoil 


"•. 

rrr  "'. 

«, 

• 

28 

41000 

-  8000 

-   11000 

O 

30 
36 

42 
46 

38000 
2*7000 
25000 
24800 

-10000 
-14000 
-  1  60  0  0 
-16000 

-14000 
-21000 
-22000 

-22000 

12600 
12*700 
12800 

48 

24700 

-15000 

-21800 

13000 

1 


Acceleration  up  plane. 
In  battery  60°  Elevation. 

v4»  .102   ve=.207 

a  -  .102 


.292 


average 


aa=.105 
.086 


v4=  .345 
a  =  .053 


a  *  .085 

.086 
ace.  » =  43  I/sec. « 


.002 


Out  of  battery  60°  Elevation 
v 


vt»6.550 


6.550  va=6.350 


ace.  = 


.050 
.133 


a  =.150 

9 


v4=6.150 
.020av.   =  .133 


13.3  I/sec. a 


.010 


(Reversed  ) 


In  battery  30°  Elevation 

v   =  .318   v  =  .475  v  «  .600   v  =  .720 
a  =  .318  a!=  .157  a~»  .125  a  =  .120 


average  *  .180  ace.  = 


180 
.002 


90  I/sec.* 


Out   of  battery   30°   Elevation 

v1=   8.630     va=   8.323      va=7*970     V4*7.600 
a  =      .307     a  =      .353     a,=    .370 

123 

.343 
average   =    .343  ace.    =    '  *     34. sec.* 

.010 
(Reversed ) 


829 


OF 


830 


Out  of  battery  30°  Elevation 

(1)  Recoiling  parts  along  bore 
i       50000-15790x.5»42100.1bs. 

(2)  Recoiling  parts  up  plane  acceleration 
-  34.  ft/sec. » 

15780 

x  34  »  16700  Ibs.  normal  comp.  16700", 91355= 

32  2 

15200.  Ibs. 

5231 
Top  carriage  up  plane  x  34  3 

32 .2 

162x34=5510. Ibs. 

5513 
Cradle  up  plane  x  34=171x34= 

33.2 
5820  Ibs. 


Stability  of  240  Caterpillar. 

Moments  taken  at  0°  Elevation,  Howitzer  out 
of  battery,  about  a  point  under  of  rear  track 
sprocket. 


(1)  Weight  of  recoiling  parts  15790x59. 
Weight  of  cradle  5231  116 

Weight  of  top  carriage        5513  80 
Weight  of  bottom  carriage      5250  45 

Weight  of  tractor  55000  128 

Inertia  of  recoiling  parts  58000  93 
Inertia  of  recoiling  parts     6700  59 

Inertia  of  cradle  10820  86 

Inertia  of  top  carriage  11580  71 


931,610 

601,565 

441,040 

236,250 

7,040,000 

5,394,000 

395,300 

930,520 
822,180 


1,703,785 
Inertia  forces  60°  Elevation  in  battery 

(1)  Along  bore  »  159000-15790  cos  30°(Recoiling  parts) 
15000-13765-145325  Ibs. 

(2)  Up  plane  »  acceleration  =  43. ft/sec.* 


831 

15790 

x  43=21070  Ibs.  normal  comp.  =  21070*. 81355 

32.2 

*  19250  Ibs.         5231 

(3)  Top  carriage  up  plane  x  123. =162. x43. =6966. Ibs. 

32.2 

5513 

(4)  Cradle  up  plane  x  12. 3=171x43. =7350  Ibs. 

32.2 

Out  of  battery  60°  Elevation 


(1)  Along  bore  recoiling  parts  =  70000-13675»56325 

(2)  Up  plane  recoiling  parts  acceleration  «  13.3  ft/sec.* 

790 

13.3  =  6517. Ibs.  normal  comp.  =  6517*. 91355 


5950.    Ibs. 


5231 


(3)  Top   carriage   up   plane  £H±  *   13.3   =    162   xis.3 

32.2 

(4)  Crldle^p'plane  |fi|  *   13. 3  =171   x   2274. Ibs. 
In  battery  30°   Elevation. 


(1)  Recoiling   parts    along   bore   147000-15790". 5=139100 

Ibs. 

(2)  Recoiling  parts  up  plane.  Acceleration  90  ft/sec. 

1 S790 

x  90=44200  Ibs.  Normal  comp.  44200*. 91355 

32.2 

=  40300  Ibs. 

(3)  T.C.  up  plane  |22I  x90=162.x90=14600.  Ibs. 

J  o  .  o 

5513 

(4)  Cradle  up  plane x90=171x90=15400  Ibs. 

32.2 

(2)  About  center  line  rearmost  roller. (llOin. from 
trunnions ) 

Weight  of  recoiling  parts  15790."  41.  647000. 
Weight  of  cradle  5231.  97.  507000. 
Weight  of  top  carriage  5513.  62.  342000. 
Weight  of  bottom  carriage  5250.  27.  142000. 
Weight  of  tractor  55000.  110.6050000. 


832 


Inertia  of  recoiling  parts  58000.  93, 

Inertia  of  recoiling  parts  6700.  59, 

Inertia  of  cradle  10820.  86, 

Inertia  of  top  carriage  11580.  71, 


-5394000, 

-  395000. 

-  930000. 

-  822000. 

+  147000 


Direct  Pads   on  Rollers. 


In  battery 

Weight  of  recoiling  parts   16700*. 99452 
Weight  of  cradle  5231x. 99452 

Weight  of  top  carriage      5513*. 99452 
Inertia  of  recoiling  parts  17500*. 99452  + 
Inertia  of  recoiling  parts!40000x. 10453  + 


Hydraulic  resistance 


58426, 


f29967-28433+140000x . 


1 


17500x.  10452-26534.  M045J 


76200. 


0°   Out  of   battery. 

Weight  - 

Inertia  of  recoil- 
ing parts 

Inertia  of  recoil- 
ing parts 


Hydraulic  resistance 


26534*. 99452   +  26389. 
6700.x. 99452  -   6663. 

58000x. 10453   +   6063. 
25789. 

J+22400   58000x.  99453      1 
1+6700X.  10452-26534.  *.  10452J 


78330 


60°  In  battery, 

Weight   15790+5231+5513    .99452   +  26389. 
Inertia  of  recoiling 

parts    (145324  x. 91355)          +  132762. 
Inertia  of  recoiling 
parts  (19250.x. 40674)  -  7838. 


166981 

[6966+73  50+19250X.  91355 
Hydraulic   resistance   <\ 

[I45325x.40674+26634x.  10453 

24400  Ibs. 


60°  Out  of  battery 

Weight  (15790+5231+5513). 99452   +    26389 
Inertia  of  recoiling  parts(56325x 

.91356)+    51456 
Inertia  of  .recoiling 

parts  (5950.x. 40674)  2420. 

75425 

&274+2155+56375X. 40674  "J 

Hydraulic  resistance  >  >-. 

|5950x .  91355-26534x  .10453] 

32800 


0°  In.       80000  Out  = 

30°  In.  Out  » 

60°  In.  Out  * 

Weight  of  bottom  carriage  5250  Ibs, 


At  30°  Elevation.  Hydraulic  resistance.   In  battery. 
13 9000*. 80902-40300*. 58779-14600- 15400-26594 

x. 10453 
114000-23700-14600-15400-2780=57520 


834 


/L 


/ 


112 


, 


z 


\ 


835 


836 


837 


"7 


\ 


838 


Bfi 
If] 

35 


i 


I 


>- 


-* — *r 


>        * 


839 


840 


841 


\ 


Fig. IS 


842 


843 


IfOO 
1OOO 

soo 
o 
soo 

IOOO 

/soo 
xooo 

24-00 

3OOO 

\ 

5  F 

C* 
VIM 

»«- 

TERPILLAR 
HOWfT7ER 

WO     HIO»     HIM 

mr  FRMMC 

VA 
f5C 

LS 

RK 

-NE 
•  »--^. 

m 

DEK 

u 

Z40 

) 

. 

- 

'   N, 

N 

• 

/ 

-; 

5t>- 

^  •* 

•  •  X 

«-   £1 

•  f.' 

tra. 

/ 

, 

/ 

/ 

i 

•     ' 

•  •    . 

-- 

' 

:.: 

. 

- 

n 

'  \  z 

3     J 

C'     « 

3     5 

3      ? 

0      ' 

0  & 

0     1 

0    /• 

•0   « 

0    /^O    /. 
INCHES 

••     ,^ 

J  .'- 

i1         ' 

0 

<0  if 

C1  / 

s  .- 

10     , 

a  ? 

Ho  t 

30  Z*0  ZfO 

, 

; 

. 

' 

! 

: 

' 

1 

\ 

HOTI 

»:-!h 
^.  P 
«n 

n  toon.  *. 

30KHCC  OP 
X  COMPRlii 

•  -» 

•s. 

.h 

r.-.   .  t 
!         M 

•1-. 

f 

IN 

H 

--+ 

1 

stf,a 

(*-  **1 

auraneo  OM 

tmcill  fr  EQUMJIEH—  \ 

* 

. 

1 

1 

, 

EC 

I 

:  3 

3 

00 
00 

i 

c 

c 

0 

c 
C 

_ 

i 

! 

> 
i 

- 

,il 

, 

, 

3 

1 

s 

H 

| 

* 

"  E 

— 
»-?«N 

-1 

A.    K 

9 

,, 

!/»1 

1 

C 
•,      - 

0 
0 

•.a.- 

FWH 

CAT 

M 

3 

5j 

§ 

oS 

L 

O 
0 

If 

i 

0         ° 

II 

f 

1 

I 

M 

5  =  TOC 

HIT 

ITEBl. 

I.     FO 

4 

^ 

ITU)  OH   VI 

<x>; 

'-  tc 

MUJ 

:• 

guuau 

f 

I 

1 

844 


845 


346 


IO 
N 

00 

L 


847 


ur> 

iZ 


848 


Hydraulic  resistance  out  of  battery 

42100*. 80902+15200*. 58779+ 5510+5820-2780 
34100*   8940  5510+   5820-2780=51590.  Ibs. 

THEORY  FOR  VARIABLE  RESISTANCE     From  the  point 
IN  UPPER  RECOIL  AND  CONSTANT  •»   by  point  method 
RESISTANCE  IN  LOWER  RECOIL       as  previously 
SYSTEM.  discussed  in  some 

detail,  we  find, 

that  the  resistance  of  the  gun  recoil  system  varies 
from  its  static  value  in  the  battery  position,  to 
very  nearly  the  recuperator  reaction  plus  the 
total  friction  of  the  upper  recoil  system,  the 
throttling  at  the  end  of  the  upper  recoil  being 
negligible  and  therefore  the  hydraulic  braking 
becoming  zero  in  the  upper  recoil  system;  further 
it  was  found  that  the  gun  recoil  braking  falls  off 
proportionally  on  the  time.  Let 

PS  =  static  braking  for  gun  recoil  system  = 
initial  braking  reaction  on  gun  recoil 

system  (Ibs) 
Pa£  *  final  or  out  of  battery  recuperator 

reaction  for  upper  recoil  system  (Ibs) 
Pf  *  final  braking  of  reaction  on  gun  recoil 

system  (Ibs) 
Rt  -  total  friction  of  upper  recoil  system 

(Ibs) 

lfr  =  weight  of  recoiling  parts  (Ibs) 
HC  =  weight  of  top  carriage  (Ibs) 
V  =  initial  upper  recoil  velocity  (ft/sec) 
Z  *  displacement  of  gun  on  carriage  (ft) 
N  *  upper  normal  reaction  between  top  car- 
riage and  inclined  plane  (Ibs) 
R  *  lower  recoil  resistance  parallel  to  in- 
clined plane  (Ibs) 

X  =  total  run  upon  inclined  plane  (ft) 
v  =  velocity  of  combined  recoil  Rel.  vsls.=0 

t1  *  prime  P0  common  recoil 


349 


The  mean  braking  zone  for  the  upper  recoiling 

parts,  becomes,     p  .p   .p 
rs   af   t 

2 

Further  the  distance  run  up  the  inclined  plane 
during  the  time  tj  was  found  to  be  approximately 
X  =  -  vitl.   The  approximate  equations  for  the 

double  recoil,  with  a  variable  re- 
sistance in  the  upper  recoil  system  and  a  constant 
resistance  for  the  lower,  become, 

ff  V   +  to  4700 
V=0.g( )  (1) 

"r 

W_   V-v  cos (0+9) 
P0-ffrsin0=  —  t ; 3  (2) 


r  viSn 
N-Wrcos(?  =  —5 

t 


(3) 


c   t 

P.cos(0+6)-Nsin(0+9)-w.sin  9-R=  --   (4) 

g  t1 

wr+wc 
X  =  1  vtf  +  -  -  v*  (5) 

2Rg 

Z  =  -  t1 
2 

DBRIYATIOK  OF  THB  PYNAMICAI.  EODATIOHS 
POINT  BY  POIMT  METHOD  COMP  UT  AT  I  OH  : 


Total  pull  between  upper  and  lower  re- 
coiling  parts: 

This  reaction  is  composed  of:-  v 

(1)  the  hydraulic  braking  pull  =Pns( )* 

vs 

(Ibs) 

(2)  the  recuperator  reaction  at  the 
relative  displacement  under  con- 
sideration —  Pa   (Ibs) 

(3)  the  friction  between  the  recoil- 
ing parts  —  Pf  (Ibs) 


850 


v«el 

Hence  P»Phs  -  +pa*pf      (Ibs) 
vs 

REACTIONS  ON  THE  UPPER  RECOILING  PARTS; 

If  Pb»  the  powder  reaction,  then  for  the  gun 
along  its  axis,  we  have, 


Pb-mr  -  -  P+Wrsin0=0        (1) 

d  v 

and  normal  to  its  axis 

d»y' 
N-mr  -—  --  Wrcos0  =0         (2) 

Integrating  equation  (1),  we  have, 

tm  Pbdt    P-W  sin0 

—  -  (  -  -  -  )  A  t=  A  vr 
l-  m 


P-W  sin0 

vn  _  vn-i  _(  -  i  -  )A  t,  vn  .  vn 
f    f        m  r    r 


bence  P-Wrsin0) 

vn  =  vn-i  +(yn_vn-i)_(  -  £  - 

r    r      f  f         m 


From  a  somewhat  different  point  of  view,  we  have 
from  (1) 


r,  ,     «,  « 
HP*W.«ini»0   since 


d  *  d  x 

~ni.. 

dv 


'  /% 

+  — —  cos(0+9)  See  acceleration  diagram, 
dt 

d*xrel   dvc 

then  Pb-nPt +  cos  (0+e)]-P  +  wrsin0+e 

dta    dt 

Integrating,    we    have 

pbdt          P-Wrsin0 

m       -(     TO  t!*vrel*vc  cos(0+9)=vr  hence,    as 

before, 

yn   _  yn~i+ (yn_yn— 1 \  _ 


mr 
Fro«  the  vector  disgram  of  acceleration, 


851 


dav'   dv 

=  sin(0+9)   hence  equation  (2)  becomes, 

dt8    dt 

dvc 
N-mr sin  (0+6)-W  Cos0  =  0      (2a) 

dt 

REACTIONS  ON  THS  LOWER  RECOILING  PARTS 

These  reactions  are  N  and  P  reversed  (the 
mutual  couple  having  no  effect  on  the  translation) 
of  the  upper  recoiling  parts,  the  braking  reaction 
K  of  the  lower  recoil  brake  and  the  weight  and 
kinetic  reactions  of  the  top  carriage. 

The  normal  reaction  and  couple  exerted  by 
the  plane  has  no  effect  on  the  motion  of  the  sys- 
tem, then,  along  the  inclined  plane, 

Pcos(0+9)-Nsin(0+9)-Wcsin  e-R-no  ~jf~  =  C 

dvc 
Substituting  N  =mr  — ; —  sin(0+9 )+Wrcos0 

01 T 

we  have      ^v 

Pcos(0+9)-m  --4  sin2  («f+6)-W_sin  6-tfrcos0sin(0+9) 
d  t  c 

-R-tDc   ••  =  0,  combining  terms  and  simplify— 
dt 

ing,  we  have 

ii£  -p 

r          c  dt  c     '  i 

hence 

dvc   Pcos(0+9)-Wcsin9-Wrsin(0+9)COs0-R 


dt       mrsin2  (0+9)+aic 

and  between  any  two  intervals, 

Pcos(0+6)-W  sin9-W_cos0sin(0+6)-R 

•j.D-Ttn-l,  r  . . 1. 

V 


GEOMETRICAL  RELATIONS, 

To  compute  P  it  is  necessary  to  compute  the 
relative  velocity  and  displacement  respectively 
for  any  given  interval  in  the  recoil.   Obviously 
from  a  velocity  diagram 


852 


vrel=vr-vcco3(0+9)  and  the  relative  displacement 
vrel~xref  +  ? A  l  and  the  displacement 

C    C 

up   the   inclined   plane   x£  =   xj}"1   +   ---•'    -  A   t 

METHOD  OF  COMPUTATIOH, 

Knowing  v°-1  ,  vp"1  and  vjjgj  at  the  beginning 
of  the  interval,  we  have, 
vn-i 

1*6 1 

P=Phs  ^ — ^"^  +pa*pf  at  relative  displacement  xg^J 
v«. 


then  v^^v"'1  +[  -  .  .  ,^  QX  -  3A  t 
c  c  m+msin2  (0+9) 


and  _ 

vnsyn-l        - 


r 


"" 


From  these  values,  we  have  v?el  =  vr"~vc 
and  therefore 

vn  +vn-l 
tfn   _  xn-i   vrel+vrel 

xrel  -   rel  +  -  -  -  A  l 
o 

and  1 


After  the  powder  period,  obviously  the  expression 
for  vr  reduces  to, 

P-Wrsin0 
m 


vnsyn-i  _  (  -  L  -  )A  t 


HELATIVB  COUNTER  RECOIL  OP  THE  UPPER 


RECOILING  PARTS: 


In  the  expression  for  P,  the  hydraulic  re- 
action and  friction  reverses.   If  B1  is  the  c're- 

coil  buffer  force  in  the  upper  recoil  system  at 
a  given  relative  displacement,  then 

v  a.  S-..-T? 


853 


Pn=  -  B'("l  —  )  i  pf=~pf  assuming  friction  the  same. 
vs 

hence,  „«  , 
y  rel 
P=Pa-B'(  -  —  )~Pf  (Ibs)   The  remaining  expressions 

are  the  same  as  before. 

This  method  of  computation  is  sufficiently 
accurate  and  was  followed  in  the  recoil  cal- 
culations illustrated. 

APPROXIMATE  CALCULATIOMS  FOB  STABILITT 
WITH  A  DOUBLE  RECOIL. 

Reactions  and  velocity  for  double  recoil 
system: 

P  =  resistance  of  gun  recoil  system 

Wr*  weight  of  recoiling  parts  (upper) 

Wc=  weight  of  top  carriage  and  cradle  (lower) 

V  =  initial  velocity 

z  =  displacement  of  gun  on  carriage 

R  *  reaction  of  lower  recoil  system 

N  =  upper  normal  reaction  between  recoiling  parts 
and  top  carriage 

M  «  lower  normal  reaction  between  top  carriage 
and  tractor. 

X  =  total  run  up  on  inclined  plane. 

v  =  velocity  of  combined  recoil 

t  =  corresponding  tine. 

0  =  angle  of  elevation  of  gun 

9  s  inclination  of  plane. 

Values  assumed  for  computation  of  recoil: 
P  * 
R  = 

9  =  6° 

Wr=  15,790  Ibs. 
«c=ll,570  Ibs. 
0  = 
General  equations  for  double  recoil: 

Wr   V-v  cos  (0+9) 

P-Hrsin0=  —   [  -  -  -  ]    (1) 
g 


854 


wr  v  sin  (9+0) 
N-Wrcos0» (2) 

g  w 

Pcos(0+9)-Nsin(0+9)-Wcsin9-R»  —  -  (3) 

[Ncos (0+9 )+Hccos9+Psin (0+9 )-M-  0]  (4) 

v    *r*we 
B  2  **   2Rg   ** 

Pz+  7(-^j^)  v*+R  |  t  »  j  mrVa  (6) 

wxv_+4700w 
V-0.9C )  (7) 

»=  |  t  (8) 

Energy  equation: 

PX0=  ^mr[vk-?acosa(0+e)]   Indication  of  P 

Nxsin(0+9)=  ~  mrva(0+9) 
t   r 

[Pcos(0+9)-Nsin(0+9)-R]x«  jMcv2 
0(X_-Xcos(0+9)]+  i  mrvasin2  (0+9)+R  x  =  -rar 

t»  z    r  I  i 

[V«-?2cos2(0+9)]-  i  m.v2 

2    l» 

a      t      Rvt    i 
hence  Pz+  -mrva+  ^v2*  — mrV2 

Further        wxvm+4700*  w 

V  »  0.9( )     where  w  =  weight 

Hr  of  shot, 

w  =  weight  of 

powder 

Wr=  weight  of  recoiling  parts 
vm  *  muzzle  velocity  of  shot 

240  M/M  DOUBLB  RECOIL  MOUNTED  OX  UARK  III  MI 
CATERPILLAR. 


SCKMZIDER  HOWITZER  AT  0*   ELEVATIOH  OF  HOWITZER. 

Given:  Wr  *  15780  Ibs.  —  weight  of  recoiling  parts 
Hc  =  11570  Ibs. —  weight  of  sliding  carriage 


855 


9=6°     —  angle  of  inclined  plane. 

V  -  45  ft/ffi  —  max.  velocity  of  upper  recoil- 
ing parts  at  beginning  of  re- 
coil. 
R  *  80000  Its.  —  resistance  to  recoil  on  lower 

recoil  system. 
From  static  force, 

Diagram  240  M/W  Howitzer, 
Ps  =  155000  max.  pull 

Rt+Paf  =60,000  maximum  recuperator  reaction  plus 
friction  at  end  of  recoil. 

Approximate  Calculations, 

Ps+Paf+Rt 
P0  »  -  -  -  whence  Ps=155,000  (Ibs) 

Paf+Rt   =  60,000  (Ibs) 

|215,000 
hence  Pc  =   107,500  Ibs.  mean  reaction 


=  1.480+0.58=  2.06  ft. 
Z  *  22.5x0.158=  3.56  ft.  =  42.7  in. 
Check  on  Z  by  energy  method: 

t,  27360,--  —  »    80,000*10.4 
107,500  Z  =  2("327F)      =  ~~^i  -  X 

-  I  1579°  Tf 
~~-  ^  32.2 

107,  500Z+46,  000+68,  200=487,  000 
Z-3.,56  ft.  Cheek 

R  =  80,000  Ibs.   For  horizontal  recoil,  -0=  0 
e=6°  sin  9  «  0.1045  cos  9  =  0.9945 
45-0.9945, 


107.500 

32.2 


856 


1S790  v 

N-15790 ~~  *  0.1045  -  (2) 

11570  v 

0,9845x107, 500-NxO. 1045-11570x0. 1045-80, 000  =— 

32 .2   t 

107, 000-0. 1045N-1210-80, 000=359  - 

v  25, 790-0. 1045N   N-15,790  L 

--   ' »  — -£ hence  7.11N=136,290 

t       ooV  DJ. .  ft 

H  »  19,170  Ibs 
51.2  v    490 (45-0. 9945v )   51.2  v 


N-!5790        107,500 

51.2v-692-15.3v 

66.5v=692  hence  v  -  10.4  ft/sec. 

51.  2x10.  f 


3380 


-1575 


Total  time  =  T+t*.  032.  158=.  190  sec. 

240  M/M  POUBH  RKCOIL  MOOHTgD  OH  MARK  IV  MI 

CATKRPILLAB. 

"~~ 
APPBOIIMAT1  CALCPLATIOHS  TOR   240  M/M  QAS-KLECTBIC 

DOOBLB  HICOIL  8TSTKM  AT   24°  1LKVATIOK  OF  HOWITZIB. 

Given.  Ir  »  15790  Ibs.  —  weight  of  recoiling  parts. 
Hc  *  11570  Ibs.  —  weight  of  sliding  carriage. 
9  »  7e        —  angle  of  inclined  plane. 
V  »  45  ft/a.  —  max.  velocity  of  upper  re- 
coiling parts  at  beginning 

of  recoil. 
R  =  45,000  Ibs.  —  resistance  to  recoil  on 

lower  recoil  system. 
From  static  force, 

Diagram  240  M/K  Howitzer 
Pg  =  155,000  max.  pull 
Rt  +Paf  *  60,000  maximum  recuperator  reaction 

plus  friction  at  end  of  recoil. 
Approximate  calculations, 


P0  »  -=  —  —  whence  Pa  «  155,000  Ibs. 


857 


P  R     -      6°'000 

"aft"   ~ 

[215,000 


107,500   Ibs. 

R  =  45,000  Ibs.,    0  *   24°  Elevation  of   gun,    6=  7° — 
angle   of   inclined   plane 
0+6=  31°  sin(0+9)=.5150  cos (0+6)= . 3572 
sin  9   =    .1219  Wrsin0=1578Qx. 515=6140  Ibs. 

107, 500=6140=490 (40"°'85y2V)  (1) 

t 

N-15,790x.8572=4SOx    >515°  V  (2) 

107, 500x.8572-N. 515-11, 570x. 1219-45, 000-359  -        (3) 

v 

359  -  =    93, 200-0. 515N-1410-45, 000 

v   _    t46, 800-0. 515    N  _    N-13520 
t   "  359  252 

46,800-0.515   N=1.425N-19250 
K  =   34,000  Ibs. 

252   v  490(45-0.8572   v) 


N-13,520  107,500-6140 

252   v        22080-420v 


SM80          101,360 

337v»   4480   hence   v   =   13.23   ft/sec. 

252x13.23 

*  *   nn    *c,n          ~   0.163   sec. 
20,480 

Total  time  T+t»0.163+.  032=0.  195  sec, 

s_  27360          13.23 

x   »  -(13.23xQ.195)    +  -     - 
4  2x32.2        45000 

x  =   1.955+1.65=3.59   ft.   »   43    (in) 

v  45 

Z  =  -  t  =  —     xQ.163-  3.67   ft.  =44   in.    check. 

6  Z 

107,500  Z   *  J(£Z££2)  I3T232    *  ^221  x   73.23   x   0.163 
32.2  2 


32.2 
74,  500+48,  500=497,  000 


858 


Z  »  3.48  ft. 

The  discrepancy  between  this  value  of  Z  of 
the  above  is  due  to  the  fact  that  work  done  by 
gravity  is  omitted  in  the  energy  equations. 


Theory  of  stability  not  braked. 

By  D'Alerobert 's  principle,  we  have 

K   *  P  —IB 
°     '   r  dt« 


(1) 
(2) 


where  KQ  *  dynamic  inertia  resistance  of  recoil- 
ing parts  *  0.9  K  (assembled  approximate) 
F  =  tractive  force  reaction 
r  *  radius  of  traction  rim 


859 


d29      d29       d29 

(F-BF)+r-Rr0»I0  TTT  *  J0  TE*  +»«!  8(7-78) 
w  v  at*  dt2 


(2) 
(3) 


r' 

(4) 


(5) 


It=2nr  mr2 

Hence  we  have  the  following  equations: 


(1) 

•v 

d2x 
(F-2F1)  r-Rr0=[I0-I0+3ar2(itr+l)}—         (2) 


r 


-^  **  (3> 

^  A  *  fl 

(4) 


f  c\ 

i-  'r2*At  W 

The  reaction  of  the  truck  rollers  on  track, 


=  40.4^2- 


i          dt« 

4 

The  reaction  of  the  clutch  shaft  pinion, 

,i.   1.75  24  20.35  dax        dax  dax 

R  (-)*  --  -  -  =  19.1  -  hence  R  »  - 

»V   1.43  5.2  6.00  dt»        dt2         »   dt» 

The  reaction  of  the  drive  gear  pinion, 

.  7.32 


860 


d*x  d*x 

/. R(0. 2165  )-(76. 4*0. 846)-— =7.32  — 4 

at*  dt 

d*x 
R  (0.2165  )-(64. 6+7. 32)—- 


R   -   — • =   332.0 

0.2165  dt* 

R  =   331.0  — -  331  .-^ 
dta  dt* 

d*x 


and    I+2mr*Ur+l) 


I0»35.54          m=       ~  =   4.65 

1^=18.5  r   =   1.43   ft.        1  =   158  in.    or   13.2   ft, 

bence  B 

35.54*9.30x1.43    (nl.  43*13.  2)  d^x        18.5  dax 

1.43  dta        1.25  dt* 


(p-40.4  Hi)i.43-331  £i  =  274.8  Hi 

dta  dt* 

663.4     d*x  ,  d*x 

bence  p   =  _._     _  =  463  — 

2l!259  +    15000  _  ^   m   838 

32.2  32.2 

d2x          d2x 
50000-463  T-T=838— 


dax        50000 

-  »  -  =   38.4   ft/sec.* 

dt*        1301 


dt         *   *   dt* 


—  )V* 
r 


R-(ir*i0k-*irkj  ~)(v 


dx 


861 


Check  on  equivalent  mass  of  rotating  parts: 

Kinetic  energy  of  rotating  parts  in  terms  of 
translatory  mass, 


+7  lO.l(^j)*  +  J-  18-5(^j)*+7  9.3(nl.43+13.2  v» 

t  28*48. 6+430 


J  10.1  Sj —   =20.2 

t  18.5 

•  \t 

82.2 


231.21 
I  Mrott 

Mrot  =  463.0 

CALCOLATIOH  OF*  STABILITY. 

Evaluation  of  inertia  couples: 

d2x 

Track  rollers  20.2  =  653 

dta 

dax 

Track  inertia  sprockets  274.8  — -  =  8900 

d  t 


Intermediate  gear  7.32  =237 

dt« 

Clutch  19.1  —=616 
dt« 

Resultant  couple  effect; 

653  10,169 

8900  237 

616  9,932  ft.lbs.  stabilizing  moment,  due  to 

119,000   ft.lbs         inertia  couples  of 

10169  wheels: 


862 


Kd=0.9K=500001bs.  Overturning  moment. 

50000x72=3, 600, 000 (  overturning  moment)lbs. 

Stabilizing  moment:  6248x69.5=2,480,000   3,520,000 
9396x111*  1,040,000     119,000 

3,520,000   3,639,000 
Mc — -  =       x  32.3  =  36400. 

36400x32.5=1,185,000 

3,639,000 
1,185,000 

4,824,000 
Dynamic 

Overturning  moment  3,600,000  Ibs. 
Stabilizing  moment  4,824,000   Ibs. 

Static 

Overturning  moment  4,100,000 
Stabilizing  moment  3,520,000 


d*x 

50000-15000-4647-*-=  1270 

d  t         d  t 

d*x        d8x 

3  500  0*1  73  4r—  T   hence  -T-T  »  20.2  ft/sec. 
at         dt' 

292x38.61 
Ft  =  M  V        t  =  -  =  0.226  sec. 


50000 
,  «0.«'.8»6  .  2.28  ft. 


S  =  S^  St  =  2.28 
.87 

3.15 


CHAPTER   XIII 
MISCELLANEOUS  PROBLEMS  AND  TYPES  OF  CARRIAGES. 

GENERAL  DYNAMIC  EQUATION  01  RECOIL     The  follon- 
DURING  POWDEF  PRESSURE  PERIOD.      ing  theory  is 

perfectly 
general  and 
specially  ap- 
plicable for  types  of  mounts  that  do  not  recoil 
along  the  axis  of  the  bore.  Let 
m  -  mass  of  projectile 
i  =»  mass  of  the  powder  charge 
v  =  absolute  velocity  of  the  shot  up  the  bore 

ft/sec. 
vx»  component  of  v  parallel  to  recoil  path 

ft/sec. 
vy=  component  of  v  normal  to  recoil  path 

ft/sec. 
vrel  *  relative  velocity  of  the  shot  in  the 

bore 

mr=  mass  of  the  recoiling  parts 
P  =  mean  powder  force 
0  =  angle  between  axis  of  bore  and  path  of 

recoil 
Ng=  normal  reaction  between  projectile 

R  =  total  resistance  of  the  recoil  system  (Ibs) 
u  =  travel  up  the  bore  (ft) 
X  =  retarded  recoil  displacement  (ft) 
Xf  =  free  recoil  displacement  (ft) 
B  =  angle  between  absolute  velocity  of  pro- 
jectile v  and  path  of  recoil. 

Assume  half  the  charge  to  move  with  the  projectile 

and  half  with  the  gun. 

The  reaction  between  the  gun  and  projectile, 

becomes  P  cos  0-Ngsin0  along  the  bore 

863 


864 


P  sin0-Ngcos0  normal  to  the  bore. 

The  equation  of  motion  of  the  recoiling  parts, 
becomes,  along  the  recoil  path, 

dV 
Pcos0-Nssin0-R=  (mr  +  0.5  ID  )  —    Integrating  and 

Pcos0-Nssin0      Rdt   dividing  by  mr, 

we  have  (      n  c-   )dt-    .  __  =  V  Now  from 
m_+0.5m       mr+0.5m 

the  vector 

diagram  of  velocities,  we  have,  adding  vectorily, 

vrej+V»v   but  since  V  =  Vj  approx.  that  is  the 

retarded  velocity  of  recoil  is  approx- 
imately equal  to  the  free  velocity  of  recoil,  we 
have  vrQi  +  Vf=  v   (approx.).  Now  in  the  free  re— 

coil   Pcos0-Nssin0 

(  -    _  -  )dt=V#   that  is  the  expression 
»r+°-5  m  Pcos0-Nsin0 


is 


v  mp+0.5m 

measured 

by  Vf  and  which  assumes,  for  given  intervals  of 
time  P  and  Ns  are  not  greatly  different  in  the 
free  recoil  as  compared  with  the  retarded  recoil. 
If  R  was  sufficiently  great  to  prevent  an  appreciable 
recoil  Ng  would  disappear  but  P  would  not  vary  even 
then  greatly  for  given  intervals  of  tine  between 
free  and  stationary  recoil.  Further  Ns  is  small 
even  in  free  recoil  as  compared  with  P,  hence  the 
above  expression  would  be  but  slightly  modified. 

Next,  considering  the  motion  of  the  projectile 
in  a  direction  parallel  to  the  recoil,  we  have 

(Pcos0-N8sin0)dt=(m+0.5)vx  but  since  *xa»relcos0-Vf 
we  have  (Pcos0-Nssin0)dt=(m+0.5m)  (vpelcos0-Vf  ) 

Combining  with  the  expression  for  free  recoil  of 

the  recoiling  parts,  (mr+0.5l)Vf3(m+0.5m)(vrejcos0-Vf  ) 

Hence,  V*  = 


cos0 


Since  8  equals  the  angle  v  makes  with  the  recoil 


865 


path,  we  also  have  (Pcos0-Nssin0)dt=(m+0.5i)vcosB 

and  therefore  (mr+0.5m)Vf>(m+0.5i)vcosB 

ffl  +  0  .  Sm       p 

hence  Vf»  m  +Q.5I         Now  B  differs  very  little 

from  0,  and  assuming  8=0 

hardly  modifies  the  recoil  effect;  further  0.5  m 
is  negligible  as  compared  with  mr.   Hence 

in  +0.5  IB 
Vj  3  -  v  cos  0  approx.   The  dynamic  equations 

of  recoil,  become 

therefore 

Pcos0-Nasin0 


m+0.5m        mr+0.5i 
(m+0,5i)vreicos0    Et 


mr+0.5i 

m+0.5l         Rt 

«  v  cos0 (approx.) 

mr          mr 

Integrating  again  X  =  fv*dt  -  - —  =  *_    cos0- 

2mr   mr+m+m 

During  the  after  effect  period  of  the  powder 
gases,  the  reaction  of  the  powder  is  approximately 
along  the  axis  of  the  bore  and  the  procedure  of 
computation  has  been  previously  discussed  in  detail. 

The  effect  of  the  reaction  Ns  is  to  deviate 
the  motion  of  the  projectile,  causing  the  projectile 
to  leave  the  muzzle  at  an  angle  somewhat  greater 
than  the  angle  0. 

To  compute  this  angle,  we  have,  v  sin  B-vre^sin0 

vrel      /Br+a*gv/m+0.5i  ,cosB 
hence  sin  B  =  — —  sin0=(      )(       ) 

v         m+0.5m  nr+0.5i  cos0 

mr+m+i  «j   mr+m+I 

tan  B=  — — —  tan0   and  B  *  tan   (————)  tan0 

mr+0.5l  mr+0.5i 

The  increase  in  ths  apparent  angle  of  elevation 
becomes  B-0  and  is  usually  small  and  may  be  neglected 
in  recoil  problems. 

On  the  other  hand,  to  compute  Ns  is  important 
since  it  causes  an  additional  load  on  the  elevat- 
ing mechanism  during  the  travel  of  the  shot  up  the  bore 


866 


HS- 


dV         ^ 

(m+O.Sro)2 

dv 

(•+0.5i)* 

mr 
*  dv 

dt 

2»r 
•  •»0.5l 

dt 

o-i  n      90(P           -i   n 

-  vm+u.  om  ;•• 

-    (approx) 
t 

•fKA 

normal  reaction  of 

the  projectile  when  a  gun  recoils  at  an  angle  (6 
with  the  axis  of  the  bore,  is  always  proportional 
to  the  powder  reaction  which  varies  from  point  to 
point  along  the  bore.  Though  the  max.  reaction 
occurs  practically  at  t~he  beginning  of  recoil, 
the  moment  is  usually  found  greatest  when  the  shot 
reaches  the  muzzle  of  the  gun. 

REACTIONS  AND  GENERAL  EQUATIONS     Consider  the 
IN  A  FECOILING  MOUNT.  recoiling  parts 

to  be  constrained 
in  movement  always 
parallel  to  the 
axis  of  the  bore,  the  constraints  being  offered 

by  suitable  guides  or  a  gun  sleeve  fixed  to  the 
cradle.   We  will  assume  rotation  possible  about 

the  axis  of  the  trunnions.  Let 

Pfrs  tbe  powder  reaction  on  the  breech  (Its) 
Q^  and  Q  =  the  front  and  rear  clip  reactions 

(Ibs) 

tan  M  =t-  the  coefficient  of  guide  friction 
Mr  and  Wr  -  mass  and  weight  of  the  recoil- 
ing parts  (Ibs) 
B  =  total  braking  force  (Ibs) 
X  and  Y  =  the  coraponents  of  the  trunnion  re- 
action parallel  and  normal  to  tbe 
axis  of  tbe  bore  (Ibs) 
£  =  elevating  gear  reaction  (Ibs) 
.)  *  distance  from  trunnion  to  line  of  action 

of  E  (ft) 

Mc  and  tfe  =  mass  and  weight  of  the  cradle  (Ibs) 
6e  -  angle  between  E  and  axis  of  bore 
vr  *  relative  velocity  of  recoiling  parts  in 
cradle  (ft/sec) 


867 


»=  angular  velocity  of  tipping  parts  about  the 
trunnion   (rad/sec) 

Ir=  moment  of  inertia  of  recoiling  parts  about 
the  center  of  gravity  of  the  recoiling 
parts. 
Itr=  moment  of  inertia  of  recoiling  parts 

about  trunnion  axis 
Itc=  moment  of  inertia  of  cradle  about  the 

trunnions. 

XQ  and  yQ  =  battery  coordinates  of  the  center 
of  gravity  of  the  recoiling  parts 
with  respect  to  the  trunnion. 

xc  and  yc  =  coordinates  of  the  center  of  gravity 
of  the  cradle  with  respect  to  the 
trunnion. 
d^b=  distance  from  trunnion  to  line  of  action 

of  B. 

T  *  /7[a+Y2  -  total  trunnion  reaction, 
r'  =  radius  of  trunnion  bearing 
nt  *  friction  angle  in  the  trunnion  bearing 
x  x  y  and  y  -  coordinates  of  the  front  and 
rear  clip  reactions  with 
respect  to  the  trunnions. 

R8ACTIOHS  OH  THE  RECOILIMG  PARTS. 

The  reactions  on  the  recoiling  parts,  consist 
of  the  reactions  of  the  cradle  QtQa  and  B,  the 
reaction  of  the  ponder  Pb  and  the  various  inertia 
forces  as  shown  in  the  diagram. 

Referring  to  fig.(l)  and  considering  the 
motion  of  the  recoiling  parts  assuming  by  D'Alemberts 
principle,  kinetic  equilibrium,  we  have 

(1)  Along  the  axis  of  the  gun 

dw         dvr 

Pb-B-(Qt+Qt)sin   u+l»rsin0-mrwa  (xo-x)-mry0 mr— =0 

d  t 

(2)  Normal   to   the  bore 

(Qf-Qt)cos   u-Hrcos0+mrway0-Br(x0-x) — +2mrwvr*0 

dt 


868 


Fig.  1 


869 


(3)     Moments  about  the  trunnion, 


dv 


r      dw 


P(e+s>Bdtb-mr—  s-I  t  —+2mrBvr(x0-x)-Wrcos0(x0-x) 


u  .xi- 

where    It=Ir+oirr»  =  Ir+mr[  (xo-x)*  +  y«    and   thus   a 
variable  with   the   recoil. 

REACTIONS  OM  THE  CRADLE. 


Referring  to  the  cradle,  we  have  the  reactions 
Q±  Q8  and  £  reversed,  of  the  recoiling  parts  on  the 

cradle,  the  trunnion  reaction  divided  into  components 
X  and  Y,  the  elevating  gear  reaction  E  and  the 

various  inertia  forces  as  shown  in  fig.(l) 
passing  through  the  center  of  gravity  of  the 
cradle,  together  with  the  inertia  cou  pie  Io  —  . 
Referring  to  fig.  (1)   we  have, 

(I1)     Along  the  direction  of  the  bore 
or  guides 

B  +  (Qt+Qt)sin  u  +Wcsin0+mcwfxc  •'nigy,,  —  +Ecos  6e-X=0 

dt 

(21)     Normal  to  the  guides, 
-(Q  -Q  )cosu-H_cos0+m,,xr  --  m,,vr2y-+Esin6  +  Y=0 

21  t»          **cj*.      B   -  ** 

at 

(3  '  )    For  moments  about  the  trunnion 
axis, 


dw 

-Wesin0.yc-Bdtb-Ej-Itc    - 


where  Itc=Ic  +  n!crc= 

EXTERNAL  REACTIOHS  ON  THE  TIPPING  PARTS 

Assuming  the  tipping  parts  to  be  balanced 
about  the  trunnions,  which  is  customary  in  order 
that  the  tipping  parts  may  be  rapidly  elevated, 

ve  have  Wrxc-Wcxc=0        mrx0-mcxc=0 

and 


and  for  the  total  weight  of  the  tipping  parts 
Wt=Wr+Wc  and  Mt=mr+mc   If  now,  we  combine, 


870 


(1)  and  (I1),  (2)  and  <2'),(3)  and  (3 ' )  and  noting 
the  above  relations,  we  will  have  for  the  kinetic 
equilibrium  of  the  tipping  parts, 

(1")     along  the  bore 

dvr 
Pb+Wtsin0+mrw*x-mp--—  +Ecos  9e-X=0 

(2")     normal  to  the  bore 
-Wtcos0-mrx  —  +  2mrwvr+Esin6Q+Y=0 

Q.  t- 

(3")     moments  about  the  trunnion 
Jvr     dw     d" 

Pb(e  +  s)-mr  -rr-  s-It— •  -ItcTT  +2mrwvr(x0-x)+Wrx  cos  0 
dt 

-Ej-Tr'sinut»0 

Therefore,  we  have  for  the  retardation,  exerted 
by  the  top  carriage  on  the  tipping  parts, 

For  the  trunnion  reactions, 


r 

d  t 

Y»Wtcos0-Esin  9e-mr(x— +2w   vr) 

For   the  elevating  gear  reaction, 

dw  ^vr 

Pb(e+s)-(It+Itc)— '  +Wrxcos^-Trlsinu1+mr[2wvr(xc-x)— -.s] 

E=  

j 

APPLICATIONS  OP  THE  PRECEEDING     When  the  brake 
FORMULAE.  cylinders  recoil 

with  the  gun  as 
in  the  slide  or 
sleigh  containing 

the  recoil  cylinders  and  rigidly  attached  to  the 
gun  used  with  the  Schneider  naterial,  the  center 
of  gravity  of  the  recoiling  parts  falls  considerably 
below  the  axis  of  the  bore.   To  offset  the  effect 
of  the  large  powder  pressure  couple  and  reduce  the 
reaction  on  the  elevating  arc,  we  may  employ  a 
counterweight  at  the  top  of  the  gun  to  raise  the 


871 


center  of  gravity  nearer  the  axis  of  the  bore  as 
was  done  on  the  155  m/m  Schneider  Howitzer  or  we 
may  introduce  a  friction  disk  at  the  elevating 
pinion,  this  allowing  rotation  of  the  tipping 
parts  about  the  trunnion. 

In  other  types  of  mounts,  a  spring  buffer 
may  be  introduced  in  the  elevating  gear  thus  re- 
ducing the  elevating  gear  to  a  small  finite  value, 
and  the  moment  effect  of  the  powder  pressure  couple 
being  distributed  over  a  longer  period. 

If  now  we  neglect,  w*x  and  2w  vr  as  small  and 
during  the  powder  pressure  period  x  being  small  we 
may  neglect  also,  x  -—  and  ffrx  cos  0.  The  re- 

actions on  the  tipping  parts,  become 

dvr 
X-Pb-ar—  +Wtsin0+G  cos  6e 

Y*Wtcos0-Esin  &6 

dw    d*r 
and  PfcCe+Bj-Tr'sinu^Ut+I^)  -  mr^~  8 


Now  Pb 

dw 


where  K=  the  total  resistance  to  recoil  during  the 
recoil  neglecting  the  rotation  effect  during  the 
powder  period. 

If  y0  is  small,  that  is  if  the  trunnions  are 
approximately  on  the  axis  of  the  bore,  we  have, 

Pb=*r  —  JL  =  R  (approx.) 
dt 

Assuming  the  brake  disk  on  the  elevating 
pinion  shaft  to  offer  a  given  torque,  we  may 
readily  compute  E.   In  other  words,  the  pinion 
bearing  is  designed  for  a  given  reaction.  This 
reaction  should  be  comparable  with  the  reaction 
required  in  the  out  of  battery  position  of  the 
recoiling  parts. 


872 


Ks+Wrb  cos0 

That  is  E  =  c( )  where  c  =  2  to  3  de- 
pending upon  max. 
allowable  angular  dis- 
placement of  tipping  parts,  where  b  =  length  of 
recoil,  K  *  resistance  to  recoil,  s  =  distance 
from  K  to  trunnions.  The  trunnion  reactions  become 

simply,  X  »K+Wtsin0+E  cos  9   Y  »  W.  cos  0-E  6 
•  t         e 

Thus  the  trunnion  reactions  are  fairly  in- 
dependent of  the  rotation  about  the  trunnions, 
being  primarily  dependent  only  on  the  elevating 
gear  reaction,  the  total  resistance  to  recoil  and 
the  weight  of  the  tipping  parts.   The  additional 


forces  induced  by  rotation  about  the  trunnion  can 
be  treated  as  secondary  forces.  ~~~~~ 

The  total  trunnion  reaction  becomes,  T=/V+Y* 

(Ibs) 

To  determine  the  angular  acceleration  with  a 
given  elevating  gear  reaction  E.  We  have,  approx. 

Pbe+Ks-Tr'ain\it-Ej'(It  +  Itc) —  hence 

dt 

Assuming     Tr1    sin  ut 

1   Et 

I 

•n 


Jt+Itc          Ks  and  Ej  as  constant, 
Ir'sin  n  +E 


since   obviously  Tr'sinut+Ej   must  be  greater  than   Ks. 

Further   since     t 
f 

Pbdt   «(m+0.5  n)v 

where   m  =  mass   of  projectile 

m  =   mass   of   powder  charge 

v  =  velocity  of  projectile   in  bore    (ft/sec) 
we   have 

(m+0.5i)ve      /Tr 'sinut+Ej-gs  ^ 

w   3   ~—^— — —  ~  (. )  t 


873 


e    (m+0.5i)ve    Tr  ^ 


where  u  =  travel  up  the  bore,  (ft)   To  allow  for 
the  reaction  effect  of  the  powder  gases,  we  will 
assume  the  free  angular  displacement  at  the  end 
of  the  powder  period,  given  by 

(ui+2m)  ve 
9j  =  "         Hence  the  angular  velocity  and 

*   tc     displacement  at  the  end  of  the 
powder  period  become 

(mv  +  4700i)     ,Tr  'sina  +Ej-Ks. 
-  e  -  (_  -  1  -  )t 


e   =    (m*2m)ue  _     Tr'simyEj-Ka        t; 

where  t  =  total  powder  period  (sec) 

v  =  muzzle  velocity  of  projectile  (ft/sec) 
u  =  travel  up  the  bore  (ft) 

The  remaining  angular  displacement  is  that  due 
to  a  constant  torque  (Tr'sinu1+Ej-Ks)  acting  on  a 
rotating  mass  with  an  initial  angular  velocity  »t. 
Hence  (Tr 'sin  u^Ej-Ks)  (6t-  9t)=  ^Ut+Itc)  "» 

and  therefore,  for  the  total  angular  displacement 

6t 

(m+2i)ue        Tr 'sin^+Ej-Ks)    t*        (It  +  Itc^wi 
tXIt  +  Itc  it^tc  2      ^   2(Tr'sinct+Ej-Ks) 

mv+4700i)          Tr'sinu^Ej-Ks 
where   w^    =( e  -( )   t. 


and  t  is  computed  by  the  methods  of  interior 

ballistics  and  T=  /x*+Y*  using  a  suitable 
value  of  E,  we  may  compute  from  the  above  ex- 
pressions the  total  angular  twist. 

GENERAL  EQUATIONS:-  ROTATION     With  rotation  of 

OF  THE  TRUNNIONS  ABOUT  A  the  trunnions  about 

FIXED  AXIS  OR  A  TRANSLATION  an  axis,  the  ele- 

OF  THE  TRUNNIONS.  vating  gear  reaction 

is  usually  reversed 
and  the  magnitude  of  the  reversed  action  on  the 


874 


**& 


Fig. 


876 


elevating  gear  depends  upon  the  product  of  the 
angular  acceleration  about  this  axis  and  the  total 
moment  of  inertia  about  the  trunnions  of  the  tipping 
parts.   Thus,  in  the  rolling  of  a  ship  or  in  the 
jump  of  a  field  carriage  where  the  angular  ac- 
celeration upon  the  mount  may  be  considerable,  and 
with  heavy  tipping  parts,  a  large  reversed  reaction 
is  exerted  on  the  elevating  gear,  which  in  turn 
modified  the  trunnion  reactions.   This  same 
phenomena  occurs  in  a  double  recoil  system,  or  in 
a  railway  mount  where  the  mount  below  the  recoiling 
parts  is  accelerated  up  in  an  inclined  plane  or 
along  the  rails. 

Let  us  first  consider,  the  angular  motion 
induced  in  the  tipping  parts  when  the  elevating 
gear  reaction  is  nil. 

Assuming  the  trunnions  to  rotate  about  an  axis 
0,  fig.  (2)   and  the  axis  of  the  bore  and  center  of 
gravity  of  the  recoiling  parts  to  be  along  the 
trunnion  axis,  then, 

The  Kinetic  reactions  on  the  tipping  parts, 

become 

(1)  The  trunnion  reaction  X  and  Y 
which  impress  the  angular  acceleration 
on  the  tipping  parts.   Due  to  the 
friction  of  the  trunnions  T  =  /X2+Y* 
has  a  moment  about  the  trunnion  axis: 

T  r1  sin  u. 

(2)  The  tangential  component  of  the 
ine,rtia  force  of  the  tipping  parts 


=  —  R  —  —  and  its  moment   about  the 

g    dt2  trunnion,  axis  becomes, 

Wt 

T  R 

W_x  r 

Let  Tm=  -  sin(0+e)  "hence  —  R1  -  (Tm)=  —  R1 
W  g     dt8       g 


dt 


876 


(3)     The  centrifugal  component  of  the 

inertia  force  of  the  tipping  parts  * 

wt    dQ 

—  R  (—  —  )•  and  its  monent  about  the 

g     dt 

trunnion  axis  becomes 


dt     g      dt 

(4)     The  rotational  inertia  couple  of 

tbe  tipping  parts 
j  _. 


where  w  =  tbe  angular 
velocity  about  the 

trunnions,  Itr=  Inonient  of  inertia  of 

the  recoiling  parts  about  tbe 

trunnions 

Itc=  moment  of  inertia  of  the  cradle 

about  the  trunnions 
(5)     The  weight  of  tbe  tipping  parts, 

its  moment  being  Wt  (TG)cos0=Wt 
W_x 

0 


(- — )cos0=Wrx  cos 
wt 


(6)  Tbe  complementary  centrifugal 
inertia  force  due  to  tbe  relative 
motion,of  the  recoiling  parts38 

2mr  *  where  x  *  tbe  relative 

d  t 

displacement  of  tbe  recoiling 
parts.   Its  moment  about  tbe 
trunnion  becomes, 

dx 

2mr(x0-x)  *  — 

(7)  Tbe  powder  reaction,  and  tbe 
relative  inertia  resistance  due 
to  tbe  relative  accelerstion  of 
tbe  recoiling  parts.   We  are 

not  concerned  with  these  reactions, 
since  their  moment  effect  is  nil, 
it  being  assumed  that  their  line 


877 


of  action  passes  through  the  trun- 
nion. 

We  have,  therefore  for  the  moment  equation 
about  the   trunnion,  considering  the  kinetic 
equilibrium  of  the  various  inertia  forces, 

Tr'sin  a  +(Itr+Itc)—  -  2mrw— -  -WrXcos0 
d  t       d  t 

Wu 
r     d2Q  r     dQ 

R'X  sin(0+e)+ —  R'X( — )«  cos(0+e)«0 

g      dt*         g      dt 

If  we  assume  R  large,  for  an  elementary  dis- 
placement, R  dQ  may  be  considered  rectangular, 

hence  the  term... 

wr     dQ 

—  R'X(-r~)*  cos(0+e)  may  be  omitted. 

g      dt 

Further  R  *  R1  appro*,   R  being  the  distance  from 
axis  0  to  the  trunnion. 

In  experiments,  conducted  by  the  French  at 
"Sevran-Livry"  the  term  2mrw  -T—  was  found  to  be 
negligible.   Hence  the  equation  of  angular 
motion  about  the  trunnion  axis  without  an 
elevating  gear  interposed  becomes, 

n   T   \dw          Wr    ^aQ 
T  r  sinal**'1tr+1tc'^fjr  -WrXcos0 RX  — T  sin(0+e)»0 

since  Trsina1  and  WrXcos£l  are  small  for  a  large 
angular  acceleration,  we  have,  approximately, 

vdw   *r 


From  this  equation  we  observe  that  immediate- 
ly upon  the  recoiling  parts  becoming  out  of 
battery,  when  the  acceleration  of  the  top 
carriage  is  backwards,  as  would  occur  in  the  jump 
of  a  field  carriage,  the  upward  rolling  of  a  ship 
or  in  the  recoil  of  the  top  carriage  in  a  double 
recoil  system  or  railway  mount,  we  have  an  angular 
acceleration  tending  to  cause  a  reversal  or  stress 
in  the  elevating  gear. 


878 


ANGULAR  ACCIL1RAT16N  OP  THI  TIPPING  PARTS. 


Invariable  Elevating  Gear  Reaction 
Introduced. 


In  this  case,  tbe  angular  acceleration  of  tbe 
tipping  parts,  is  the  same  as  the  angular  ac- 
celeration of  the  system  about  the  fixed  or  in- 
stantaneous axis  0.   To  impress  this  angular  ac- 
celeration on  the  tipping  parts,  as  would  occur  in 
the  jump  of  a  field  carriage,  or  in  the  upward 
rolling  of  a  ship,  tbe  elevating  gear  reaction  is 
lessened  or  completely  reversed  when  tbe  trunnions 
are  located  along  tbe  bore.  Considering  fig. (3) 

Let  Pfc  =  the  powder  reaction  on  tbe  breecb  (Ibs) 
Qt  and  Qa  =  tbe  front  and  rear  clip  reactions 
(Ibs) 

tan  u  =  f  =  coefficient  of  guide  friction 
mr  and  wr  =  mass  and  weight  of  recoiling  parts 
(Ibs) 

8  =  total  braking  force  (Ibs) 

X  and  Y  =  components  of  tbe  trunnion  reaction 

parallel  and  normal  to  tbe  bore  (Ibs) 

E  =  elevating  gear  reaction  (Ibs) 

j  =  distance  from  trunnion  axis  to  line  of 

action  of  E  (ft) 

9e  =  angle  between  E  and  tbe  axis  of  tbe  bore 
7r  =  relative  velocity  of  recoiling  parts  in 
cradle  (ft/seo) 

dQ 

—  =  angular  velocity  impressed  on  tipping  parts 

dt    (rad/sec) 

Ir  -  moment  of  inertia  of  recoiling  parts  about 

center  of  gravity  of  recoiling  parts 
Itr  moment  of  inertia  of  recoiling  parts  about 

trunnion  axis. 
Itc  =  moment  of  inertia  of  the  cradle  about 

the  trunnion  axis. 
xo  and  yo  =  battery  coordinates  of  the  center  of 

gravity  of  the  cradle  with  respect  to  the 


879 


Fig.  3 


880 


truonioD  axis, 

x   and  yt  =  battery  coordinates  of  the  center 
of  gravity  of  the  cradle  with 
respect  to  the  trunnion  axis. 
dtb  -  distance  from  trunnion  axis  to  line  of 

action  of  B. 

r'  =  radius  of  the  trunnion  bearing, 
u  =  friction  angle  in  the  trunnion  bearing. 
x  y  and  x   y   =  coordinates  of  the  front 

and  rear  clip  reactions 
with  respect  to  the 

trunnions. 

BBACTIOM6  OH  THE  RECOILING  PARTS. 

The  reactions  on  the  recoiling  parts  are: 

(1)  The  powder  force  —  Pb  (Ibs) 

(2)  The  reactions  due  to  the  con- 
straint of  the  cradle  —  Q  and  Q2 

(Ibs) 

(3)  The  braking  force  exerted  by  the 
cradle  —  B  (Ibs) 

(4)  The  relative  tangential  inertia 

force  due  to  the  relative  acceleration 
dv        of  the  recoiling  parts  — 
mr  -^  — (Ibs) 

(5)  The  relative  complementary 
centrifugal  force  due  to  the  com- 
bined angular  and  relative  motion  of 
the  recoiling  parts  

dQ 
2mr  vr  —  (Ibs) 

d*  (6)     The  tangential  inertia  force  due 

t.       rotation  a"bout  the  axis  0 

m.R  (Ibs) 

dta 

(7)  The  centrifugal  inertia  force 
due  to  rotation  about  the  axis  0- — 

»rR(^)2  (Ibs) 

(8)  The  weight  of  the  recoiling 
parts  Wr  (Ibs) 


881 


(9)     The  angular  couple  resisting 

d*Q 
angular  acceleration  Ir—-j  (ft.lbs) 

d  * 

Tbe  equations   of  notion  for   the   recoiling 
parts,   become,   along  x  x1  — 


avr  d*Q  dQ 

pb-ar  — mr  R  -777-  cos(e+0)-MrR(T-)     sin(e+0)+Wrsin0 


since  u  =  0  (1) 

along  v  v ' 

dQ    ,/dGN,          r,  d'Q 
rv_  -—  +mrR(— --)zcos(e+0)-mrR  -— 
dt     at  at 

(2) 

for  moments  about  the  axis  0,  we  have, 

dvr 

r  dt 

.   d«Q     .   d«Q  dQ   . 

-ffi-R* 1_ +2m_vr —  [xx-x+Rsin(£J+e)] 

dt9          dt»  dt 


-Wr[ (x0-x)cos0+Rsin  e- 

Now  mrR2+Ir=Ior  moment  of  inertia  of  recoil- 
ing parts  about  axis  0. 
Hence  the  above  expression  reduces  to, 

(Pb-mr^)[Rcos(e+0)+s]+Pbeb-ZW0(Qt  +  Q2+B)-Ior  j^~ 
a  t 

dQ 

+2mr»r  —  [x0-x+Rsin(0+e)]-Wr[^0-x)cos£!+Rsin  e-yosin0] 
d  t 

(3) 

BBACTIOHS  OH  THB  CRAPLB; 

The  reactions  on  the  cradle  are: 

U)     Tbe  reactions  of  the  recoiling 
parts  on  the  cradle  Q4Qa  and  E. 


(2)     The  trunnion  reaction  T  =  /X2+Y2 

and  having  a  moment  about  its 
center  line  Tr1  sin  u. 


882 


(3;     The  elevating  gear  reaction  E 

(4)  The  tangential  inertia  force 

.  R  *I£ 

dt*  da  , 

(5)  The  centrifugal  inertia  ra«(—  )2 


c 
dt 


(6)     Tbe  weight  of  the  cradle  Wc 
The  equations  of  motion,  become 

along  the  x  x1  axis 


(Q.  +Qa)sin   u   +B+»Tcsin0-t-Ecos   9 

d*Q  .dQ 

—  cos(e+0)-mcR(— 


d*Q  dQ  a 

-mcR  —  cos(e+0)-mcR(— )    sin(e+0)-X  =0        (I1) 


along  the  y  y1  axis 

Y+Esine  -W  cosD-CQ  -Q  )cos  u-nn_R(—  )a  cos(e+0)   (21) 

d  t 

for  moments  about  the  axis  0, 

d*0 
2M0(Qt*Q2*B)-XR  cos(0+e)+YR  sin(0+e  )-mcRa  - 

dt2 

d*Q 
-Iv—  >  +Ecos6e[R  cos(0+e)-Jcos9e]+Esin9e[Rsin(e+0)-J 

sin9e)-Wc(Rsin  e-xccos0+ycsin0)=0 
Now,«cRa+Ic=Ioc  the  moment  of  inertia  about  the 

axis  0  of  the  cradle. 
and  Ecos9etRcos(0+e)-Jcosee]-»-Esinee[Rsin(0+e)-Jsin  Qe] 

=ER  cose  cos(0+e)+ER  sin0esin(0+e)-EJ(co3aee+sin*ee) 

=ERcos(0+e-9e)-BJ=E[Rcos(£)+e-ee)-J] 

Hence  the  moment  equation  of  the  cradle  about  0, 


reduces   to  ZM   (Q  +Q  +B)-XRcos  (0+e)  +  YR  sin(0l+e)-Ioc—  - 

dt* 

+E[Rcos(0+e-ee)-J]-Wc(R3ine-xccos0+ycsin£J)=0   (3  '  ) 

HgACTIONS  ON  THE  TIPPIMQ  PASTS. 

Since  the  tipping  parts  are  balanced  about  the 


883 


trunnions  in  the  battery  position,  we  have, 
and 


*rxo~wcxc*°       •rxo~moxc=0 


Adding  (1)  and  (I1),  (2)  and  (21)  and  (3)  and  (3»), 
we  have 

d*Q  dd  t 


(Hr+Wc)sin0+Eeos8e-X=0  (1") 

dQ  jn   - 

Y+Esine«»-(Wr+W..  )cos0+2mrvr-— +  (mr+nu 
c          r      c  r    r  j  *     *    r      c 

d*Q 
-(•r-t-mc)R— — •  sin(0+e)=0 

dvr 

Cl  If  M    W  V_l    U 

[x0+x+RCZJ+e)]+Wrxcos0-(Wr+Wc)Rsin  e-XRcos  (0+e) 

+YR  sin(0+e)   +  E  [Rcos(0+e^-9e)-J]    =0  (3") 

Equations  (1"),  (2"), (3")  are  the  general 
equations  of  a  recoiling  system,  where  the 
relative  translation  is  along  the  axis  of  the 
bore  and  the  trunnions  have  a  rotation  about  some 
fixed  axis  0. 

These  equations  may  be  simplified  as  fol- 
lows: 

W^aWr+Wc  ana  mt=iBr+mc  where  Wt=  the  total 

weight  of  the 
tipping  parts. 
mt=  the  total  mass 

of  the  tipping 
parts 
Further  Ior»Itr+mrR* 

Ioc*rtc*"cR*  and  Ior+Ioc=tItr+Itc+BtRt 
Rcos  (0+e)+s*Rcos(0«-e)  approx. 

dvr       d2Q         dQ 

Xapw~mr mt  Rt cos(0+e)  +  ( — )* 

b   dt        dt*         dt 


884 


d*Q         dQ  dQ 

[  -  sin(0+e)-(  —  )*cos  (j0+e)]-2mpvr—  '  -Esin0ft 
dt«         dt  dt 


*Dd  d«Q       dvr 

-(Pb-ar— 


Rcos(0+e-ee)-J 
sin  ee-Wpxcos0+HtRsin  e+XRcos(0+e)-Y  in(0+e) 

1R) 

Substituting  the  values  of  X  and  Y  in  the  equation 
of  £  and  simplifying,  wa  have, 


J 

(4) 

which  is  evidently  the  moment  equation  of  the 
various   kinetic  reactions  on  the  tipping  parts 
about  the  trunnion  as  an  axis.  Since  the  term 

dQ 
2mrvr—  (x0-x)  is  always  small,  the  elevating 

^*  dv     gear  reaction,  reducesato 

Itr  +  Itc)—    (4') 


where  Itr*Ir+«r[  (xo-x)8+y«l 

Ir=  JDoment  of  inertia  about  center  of  gravity 
of  recoiling  parts.  Hsnce  I^r  is  a 
variable  depending  upon  the  displace- 
ment in  the  recoil  x,  also  Itc=Ic*mc  ^xc+y 

a  constant 
Ic=  moment  of  inertia  about  center  of 

gravity  of  ths  cradle. 

The  equation  (4)  or  (41)  is  of  special  im- 

portance in  the  study  of  the  variation  of  the 
elevation  gear  reaction.  The  angular  acceleration 


be  detericined  in  the  following  discussion 


886 


on  the  jump  of  a  carriage. 

In  the  case  when  s  and  eb  =  0,  that  is  when  the 

center  of  gravity  of  the  recoiling  parts  and  trun- 
nion axis  lie  along  the  axis  of  the  bore,  we  have 

-(Ti  +T*  )  -     Thus  the  elevating  gear  reaction 


dt8 


is  reversed  and  its  monent 


about  the  trunnion  imparts  the 
required  angular  momentum  in  the  tipping  parts. 

We  calculate  the  value  of  (-E)  we  must  determine 

d2Q 

the  maximum  angular  acceleration  —  -—  . 

d  t 

The  condition  that  there  will  be  no  reversal 
of  stress  on  the  elevating  gear  on  the  jump  of 
a  field  carriage,  is  that 


Wrxcos0+(Pb-mr—  )s+Pbeb      rc 

dvr 
Now  roughly  Pb~flr  "~  ~~~  ^  tne  static  resist- 

dt 

ance  to  recoil,  hence  for  no  reversal  of  stress 
on  the  elevating  gear, 

In  the  battery  position: 

Ks+Pbeb  ^  (itr+itc 

Cut  of  battery  position: 


>/       .d2Q 
cos  0  =(It.r  +  It.« 


From  these  equations  we  may  determine  the 
required  distance  from  the  center  of  gravity 
of  the  recoil  parts  to  the  trunnion  axis,  to 
prevent  a  reversal  of  stress  on  the  elevating 
gear  when  the  gun  jumps  as  in  a  field  carriage. 

RiCTILIKBAR  ACCELERATION  OF  THE 
TIPPINS  PASTS 

With  a  double  recoil  system,  or  in  the  case 
of  a  railway  recoiling  along  the  rails,  the  trunnions 
are  accelerated  to  the  rear  due  to  the  recoil  re- 


886 


Fig.  A- 


887 


action  of  the  gun.  Thus  the  tipping  parts  are  sub- 
jected to  a  rectilinear  acceleration  to  the  rear 
and  the  elevating  gear  reactions  is  increased. 
Considering  fig.(  4)  we  have  the  various  re- 
actions as  the  recoiling  parts  and  cradle  as 
shown. 

BKACTIOHS  OH  THE  BgCOILIHQ  PARTS. 

The  reactions  on  the  recoiling  parts,  consist, 

(1)  The  powder  force  --  Pb   (Ibs) 

(2)  The  reactions  due  to  the  constraint 
of  the  cradle  ---  Ot  and  Qf   (Ibs) 

(3)  The  braking  force  exerted  by  the 
cradle  —  B  (Ibs) 

(4)  The  weight  of  the  recoiling  parts  — 
Wr   (Ibs) 

(5)  The  kinetic  reaction  of  the  recoil- 
ing parts  due  to  the  relative  accel- 

eration —  dv_ 

«r—  -   (Ibs) 

(6)  The  kinetic  reaction  due  to  the  ac- 
celeration  <Jy 

(—  —  )  of  the  constraint  of 

d*  the  recoiling  parts, 
i.e.  the  top  carriage  and  cradle  — 

dvc 

•r  —  (lb" 

Then,  for  the  kinetic  equilibrium  of  the 

recoiling  parts,  along  the  axis  of  the  bore 

dvr     dvc 

pb-or  --  "i-—  cos(»+»)+Wrsin«J-(Q1+0J[)sin  u  -  B=0 
dt      dt 

along  the  normal  axis  to  the  bore, 

dvc 

(Qz-Qi)cos  u-Wrcos0-mr  -  sin(0+a)=0   (2) 

d  t 

and  for  moments  about  the  trunnion  axis, 

u 


dvr  dvc 
-mr[  -  +  -  cosd^+a^lttQ-WpCos   0(xQ-x) 

u  v   u.  v 


888 
d? 


r— -  sin(0+a)(x0-x)+Wrsin0y0  =0      (3) 


dt 

For  the  kinetic  equilibrium  of  the  cradle, 
ire  have,  along  the  axis  of  the  bore  or  guides, 
(Qt+Qa)sin  u+E  cos  QQ+B-mQ—~  cos  (0+a)+Wcsin0-X=0 


along    the   normal   to   the  guides   or  bore, 

dvc 
Y+Esin8a~(Q2"~Qt  )cos   u  -Wccos0— mc— r— sin  (fif  +  a)=0        (21) 

and   for  moments    about   the   axis   of   the   trunnions, 

dvc 
it      a    t   COS1          iyt      eyz    s  b    mc   (jtcos       ra   yc 

dvc 
+mc— -  xcsin(0+a)+wccos0xc-Wcsin0yc-Ej  =  0  (31) 

dt 

REACTIONS     ON     THE     TIPPING    PARTS. 


Since  the  tipping  parts  are  usually  balanced 
about  the  trunnions  in  the  battery  position,  we 

have, 

wrxo~wcxcs°  "r^c"  mcxc   =0 

and 


Adding  equations  (1)  and  (I1),  (2)  and  (2'),  (3) 
and  (3'),   we  have,  then,  along  the  axis  of  the 

bore,dVr        dvc 

Ph-m_  --  (m_+m_)—  -  cos(0+a)  +(W,+W_  )sin0+Ecos  0a- 
"  dt    "  c  dt 

d") 

along  the  normal  to  the  axis  of  the  bore, 

dvc 
Y+Esin9Q-(Wr+Wc)cos0-(mr-nnc)—  -  -sin(0*a)»0     (2") 

dt 
and  for  moments  about  the  trunnion, 


dv  dv 


c 


.— •  sin((2f+a).x-Ej=0 
The  elevating   gear   reaction,   becomes, 


889 


dvr  d»e 

(Pb-mr-—-)y0+Pbab+Wrx  cos0+mr--—  sin(Gf+a).x 

d  t  d  t 


From  equation  (1), 
dvr   dvc 

PK-BP[—  rr  --  TTCOs(0+a)]  +Wrsin0-(0  +Q,  )sin  u-B=0 
dt    at  *   a 

Since  the  displacement  and  velocity  of  the 
top  carriage  is  small  at  the  beginning  of  recoil, 
the  relation  vr=  the  static  velocity  vs,  that  is 
the  velocity  of  the  recoiling  parts  when  the  top 
carriage  is  assumed  stationary.        cv* 

The  braking  force  B,  equals,  B=F?+-^j— 

"x 
but  since  vr-vg  approx. 

cv£     cv£ 

B=F»  »ag"sF..+  T^  ~  B-     where  B_=  the  static  re- 
v  wx   v   w* 

coil  braking  force. 

Hence  the  kinetic  reaction  along  the  bore,  becomes, 

dvr  dvc 
Pb-mr[    I    cos(0-»a)]  =  BS  +  (Q  +Q  )sin  u  -Wrsin0 

CL  if   Cl  ti 

But  for  the  static  resistance  to  recoil,  we  have 


)sin  u-Wrsin 


hence 


dv,  dv, 


X    C 

Pb-mr[  -  +  -  cos(0+a)]=Kx   Therefore,  the 

dt  dt  elevating  gear  re- 

action reduces  to, 

dv 


This  equation  is  of  special  interest  since 
in  the  battery  position,  we  find, 

dvc 
pb«b   *Kxy0+mr—  y0cos(2>+a) 


E=  when  the  top  car- 

J  riage  moves. 

0 

when  the  top  carriage  is  stationary, 


J 
Thus  we  have  only  a  slightly  additional  load 


890 


mr— 

brought  on  the  elevating  gear  (Ibs) 

J 
This  value  however  is  somewhat  compensated 

by  the  slightly  decreased  value  of  Kx  due  to  the 
fact  that  the  relative  velocity  is  somewhat  less 
than  the  static  velocity  of  recoil. 

RECAPITULATION. 


Reaction  of  Top  Carriage  on  Tipping  Parts: 
For  tbe  Trunnion  Reactions 


dvr        dvc 

=Pv-inr--  --  (mP+nic)---rcos(0+a)+(Wr-»-Wc)sin0+Bcos9e 
0  *  dt         dt 

dvc 
m+m)  -  sin(0+a)-£  sin  9 


dt 

For  the  Elevating  Gear  Reaction 
dvr  dvc 

in  (0+a) 


J 
If  we  define  Kx=B+(Qt+Q2)sin  u+Wrsin0 

then     dvr   dvc 

Kx~pbsi"r[  -  +  -  cos(0+a)]   and  Wt=Wr+Wc  total 

*  weight 
of  the  tipping  parts,  Mt=mr=mc  Total  mass  of 

tipping  parts. 
For  the  Trunnion  Reactions 

dvc 
X=Kx=mc-—  —  cos(0+a)+Wtsin0-«-Ecos9e 

dvc 

Y  sW^cos^+mt-—  sin(0+a)-B  sin9.» 
at  c 

For   tbe   Elevating   Gear  Reaction: 

dvc 
Pbeb*Kxyo+wrx   cosef+mr—  [xsin(8f+a)+yocos  (0+a)] 

_  ___^_  __  dt 

E=^  -  ' 

J 

ON  THE  JUMP  OF  A  FIELD  CARRIAGE     Mounts  are 

frequently  de- 

signed for  stability  at  a  given  minimum  elevation 
and  yet  may  be  fired  at  a  lower  elevation.  Con- 


891 


sideration,  therefore,  must  be  given  to  the  inertia 
loadings  and  corresponding  reactions  induced  by  the 
jump  of  the  carriage.   ID  the  following  discussion 
it  will  be  assumed  the  total  mount  to  rotate  about 
its  spade  point. 

By  the  application  of  D1  Alerabert  's  principle 
we  introduce  the  various  inertia  effects  as  kinetic 
reactions,  the  mutual  reactions  between  the  parts, 
of  course  having  no  effect  on  the  kinetic  equilibrium 
of  the  total  system,  gun  cradle  and  carriage. 

From  the  acceleration  diagram  we  have  for  the 
recoiling  parts, 

(1)  The  relative  acceleration  along  the 
axis  of  the  bore  -- 

dvr 

—  Ut/sec.«) 
dt 

(2)  The  tangential  acceleration  of'tbe 
recoiling  parts  about  the  axis  0  - 

BS= 

dt 

(3)  The  centripetal  acceleration  of  the 
recoiling  parts  towards  the  axis  0  — 
w«R 

(4)  The  acceleration  due  to  the  relative 
motion  combined  with  the  rotation 

of  the  recoiling  parts  2w  vr 

The  accelerations  in  the  remainder  of  the  mount, 
the  carriage  proper,  become 

(1)  The  tangential  acceleration  —  kc""~~ 

dt 

(2)  The  centripetal  acceleration  —  w2Lc 

KIHSTIC  EQUILIBRIUM  OP  THE  SYSTEM. 


(Gun  and  Carriage) 

Prom  the  principle  of  D'Alenbert,  we  have  the 
external  reactions  in  equilibrium  with  the  various 
kinetic  reactions  induced  by  the  angular  rotation 
of  the  mount  and  the  relative  acceleration  of  the 
gun. 

The  forces  and  kinetic  reactions  on  the  system 


892 


gun  and  carriage  are  : 

(1)  The  total  powder  reaction  P^ 

(2)  The  weights  of  the  recoiling 
parts  and  carriage  W^  and  Wc  (Ibs) 

(3)  The  tangential  inertia  force  of 
the  recoiling  parts  due  to  the 
angular  acceleration  about  the 

spade  point  0  

dw 
MRR  —     (Ibs) 

(4)  The  centrifugal  inertia  force  of 
the  recoiling  parts  due  to  the 
angular  velocity  about  the  spade 
point  0  

MR  R  w2  (Ibs) 

(5)  The  inertia  resistance  due  to  the 
relatire  acceleration  of  the  recoil- 
ing parts 

(Ibs) 


dvr 


r  dt 

(6)  The  inertia  resistance  due  to 
the  combined  rotation  of  the 
recoiling  parts 

2  mrwvr  (Ibs) 

(7)  The  tangential  inertia  force 

of  the  carnage  proper  due  to 

the  angular  acceleration  about  the 

spade  point  0  

dw    .   . 
fflc  c  dt 

(8)  The  centrifugal  inertia  force  due 
to  the  angular  velocity  about  the 
spade  point  0  mo  Lcw*   (Ibs) 

(9)  The  inertia  couple  about  the 
center  of  gravity  of  the  recoiling 
parts  due  to  the  angular  acceleration 
of  the  system  — 

IT?  TT   (ft. Ibs) 


893 


(10)     The  inertia  couple  about  the 
center  of  gravity  of  the  car- 
riage proper  due  to  the  angular 
acceleration  of  the  system  — 

*£$*$•  'oJi   (fulbs)  <r»«' 

For  moments  about  the  axis  0,  we  have, 


dvi 


dw 


P    (d+e)-(DR  d-mRR 

dt  dt 


(x0-x)-Wp[(x0-x)cos0-d   sin0] 
dw  -a   dw  dw 

*4t  smcLc    dt     c  dt     c  c  cos  <e-B>  -  0(1  ) 

n 
since  9  =  -%  +0+Q  whence  Q  =  angle  turned  in  rotating 

about  0,  we  have  ^Q   ^ 

—  =  —  =  w   for  the  angular 
dt   dt 

velocity 

dae   daQ   dw 

-^—  =     =  —  for  the  angular  acceleration. 

dt*   dt2   dt 

Considering  now  the  recoiling  parts,  above, 

we  have 

d  v  t?       d  w 
pb~aR mR  d  —  -mR«»*(x0-x)-B-Rt+Wrsine(=0. 

dt        dt 

R     dw 
Simplifying  we  have,  Pb^Rt-jT  +  d  T^-«-w^c0-x)]-B-Rt 

+Wrsin(?=0  (2) 

whence  B=Fv+Pb  —  tne  total  braking  reaction  (Ibs) 
Py  =  the  recuperator  reaction  (Ibs) 

Pn  -  the  total  hydraulic  resistance  (Ibs) 
Now        v* 

pb=phs  ~     whence 

vg  =  static  recoil  velocity  (ft/sec) 

Pns  =  corresponding  static  hydraulic  braking 
reaction  (Ibs) 


894 

a 
dvR     (jw  VR 

We   thus   see   that  Pb~IBRtT7-fdTrtw*  (xo~x)33phs  ~T~*Rt 

dt   ut  v 

Fv-WRsin0  (3) 

From  equation  (1),  we  have 

dvp 

Pb(d+e)-mR  -  d+2mRw  vR(xo-x)-WR[  (xQ-x)cos0-d 
d«   _  _  dt 

dt   "  mRR*-mcL0=IR+Ic 

sin£J]-WcLc  cos(0-B) 

If    Is3    the   moment   of   inertia  of   the   system  about 
the   axis    0,    then   Is»mRR*+mcLc+IR  +  Ic      a  variable 

since  mRR*«mR[da+(xo-x  )a]      a  function  of   x,    hence 

dvR 
pb(d+e)~"mR~d"[  d+2mRw  vR(xo-x)-WR[  (xo-x)cos(Z(-d 

dt  5  Is 

sin]-W.Lrcos(6-B) 

-  —  -  (4a) 

Substituting   the   value  —  in  equation    (3),    we    have 

dt       dvR 

a  dynamical  equation  in  terms  of  —  —  —  and  w.   If 

dt 

now.  we  construct  a  table  for  the  various  intervals 


of  time,  we  may  compute  VR,   p  ,  w  and  J*  by  the 

d  t        dt 

methods  of  a  point  by  point  procedure. 

APPROXIMATE  CALCULATIONS  FOR  THE     Prom  equation 
JUMP  OF  A  CARRIAGE  (3)  in  the 

previous  article, 
we  have 

dvR       dw  VR 

Pb-mR[  -  +d.  —  +v»2  (x0-x)]-Phs  —  -vFv+R 
dt          dt  v 


— 

Cl  tr 


The  terms  m  d—  and  mR  w2  (XQ-X)  are  usually 


small  compared  with      dvR 


dvi 


mR  ,    hence   VR=VS   approx 


dt 


and  Pb~mR~j~  =  ^  ^^e  statlc  resistance  to  recoil 

(approx ) 


895 


COMPONfA/TS  OW  JUMP 


/WffTM  SOffCfS  ON  JUMP  OF  F/fLD 


Fig.  5 


886 


Substituting  in  equation  (4a)  of  the  previous  article, 
and  omitting  the  term  2  mr«r  vr(xQ-x)  which  is  small, 
we  have 
dw   Pbeb+Kd-Wr[(x0-X)cos  0-d  sin0]-W0Lc  cos(e-B) 


The  moment  effect  of  the  weights, 
Hp[(x0-x)cos0-d  sin£J)=  WcLccos  (9-B)=  wsLs~BR 
Hence  dw   Pbeb=Kd-WgLg  +WRx  cos  0 

**  "    "V 
where  Ws  =  weight  of  entire  system 

Ls  =  horizontal  distance  from  spade  point  to 

line  of  action,  of  Ws 
Is  =  moment  of  inertia  of  total  system  about 

spade  axis 

BARBETTE  CHASSIS  MOUNTS.     In  this  type  of  mount, 

the  top  carriage  and  gun 

recoil  up  an  inclined 
plane,  and  the  recoil 
in  general  is  not  parallel 
to  the  bore. 

The  characteristics  of  such  mounts  is  that  a 
component  of  the  direct  powder  reaction  is  brought 
upon  the  mount  and  therefore  the  various  parts  are 
stressed  considerably  higher  then  with  mounts  recoil- 
ing in  a  cradle.   During  the  powder  period,  we  have 
an  impulsive  or  percussion  effect  brought  on  to  the 
mount,  and  the  effect  of  finite  forces  as  gravity 
and  the  braking  force  may  be  neglected. 

The  gun  together  with  the  top  carriage  are 
considered  in  this  type  of  mount  as  the  recoil- 
ing parts.   The  gun  has  trunnions,  and  the  trunnions 
are  located  at  the  center  of  gravity  of  the  gun 
along  the  axis  of  the  bore.  Since  there  is  no 
regular  acceleration  in  the  recoil,  the  reaction 
on  the  elevating  gear  is  practically  nil.   Due  to 
the  weight  and  position  of  the  center  of  gravity 


897 


ON  fffCO/L/A/G 


Fig.  6 


898 


of  the  top  carriage,  the  center  of  gravity  of  the 
recoiling  parts  is  not  located  at  the  axis  of  the 
bore.   During  the  powder  pressure  period,  there- 
fore, we  have  a  whipping  action  due  to  the  powder 
pressure  couple  which  increases  the  end  roller  re- 
action and  the  front  clip  reaction. 

The  bottom  carriage  which  supports  the 
chassis  for  the  top  carriage  is  traversed  on  a  roller 
base  plate,  the  horizontal  reaction  being  carried 
on  the  pintle  bearing  and  the  vertical  reactions 
by  the  traversing  rollers.   This  arrangement  is 
typical  of  any  Barbette  emplacement.   Let 
Fb  -  the  total  powder  reaction  (Ibs) 
0  =  angle  of  elevation  of  gun 
a  =  angle  of  inclination  of  chassis 
fflg  and  Wg  =  mass  and  weight  of  the  gun  (Ibs) 
DC  and  wc  3  mass  and  weight  of  the  top  carriage 
mr  and  wr  -  mass  and  weight  of  the  recoiling 

parts 

8  =  the  total  braking  reaction  (Ibs) 
dt>=  distance  from  trunnion  to  line  of  action 

of  B  (ft) 

QA  and  0^  =  the  front  and  rear  roller  reactions 
on  the  top  carriage  exerted  by 
the  chassis  (Ibs) 

Rt  and  Rf  =  the  front  clip  reaction  and  rear 
roller  reaction  exerted  by  the 
traversing  base  plate  on  the  bottom 
carriage   (Ibs) 
dx  and  dt  =  distance  from  trunnions  to  line 

of  action  of  Q  and  Q  respectively, 
n  =  friction  angle  of  roller  reactions. 

H  -  the  horizontal  reaction  between  the  base 
plate  and  bottom  carriage  at  the  pintle 
bearing  (Ibs) 

REACTIONS  OtT  THE  RBCOILIHG  PARTS 

GUN  AMD  TOP  CARRIAGE  T06STHBR 

We  have  for  the  motion  of  the  recoiling  parts, 
along  the  chassis:- 


899 


d*l 

Pbcos(0+a)-W_sin  a-B(Q  +Q  )sin  u-m.  —  =•  =0       (1) 

rdt» 

normal  to  the  chassis:- 

PhSin(0+a)+Wrcos  a-(Qt+Qt)cos  a  =0  (2) 

about  the  trunnions:— 
d*l 


If  we  assume  the  braking  constant  throughout  the 
recoil,  we  have,  B+Wrsina+(Qi+Qa  )sin  u  =  K  and 
equation  (1)  becomes, 

Pbcos(0+a)-mr—  •  -  K  =0 

Integrating,  we  find, 

dl        Pbcos(0+a)      K 

—  =  v  =  /  -  dt  --  t 
dt  m          m 


=  Vf  or  the  maximum  free 

r        r 

velocity  of  recoil  for 
a  recoiling  mass  mr,  hence 

Kt 


t 
dt 

Integrating  again,       T  Kp 

1.  *  /   Vfcos(0+a)dt-  — 
o  2mr 

KT2 

=  E  cos(0+a)  -  —  •  where  E  is  the  free  recoil  dis- 
2nir 

placement  for  a  recoiling  mass 
m.  during  the  total  powder  period.   During  tfte  re- 

g 

mainder  of  the  recoil,  we  have  J»rv*=K(b-lt  )  hence 

rr  t  f^  flpg 

-mr[Vfcos(0+a)  --  ]*  *K[b-Ecos(0+a)+  —  ] 
mr  2mr 

Simplifying,      mfV|  cos«(cr+a) 

K  =  -  where  E  and  T 
2[b-(E-VfT)cos(0+a)]  are  obtained 

by  the  methods  of  Interior  Ballistics. 

EFFECT  Of  CHASSIS  ROLLER  REACTIONS  OH  THE  RKCOIL 
BRAKE. 

Assuming  only  the  end  roller  reactions  to  come 
into  play,  we  have,  from  eq.  (1)  and  (2), 


900 


K-B-Wrsin  a 


tan  u  =  •  =  f 

Pbsin(0+a)+Wrcos  a 

hence  K  -B-Wrsina=f [Pbsin(0+a)+Wrcos  a]  where  f  » 
coefficient  of  roller  friction.   After  the  powder 
period,  K-B-Wrsin  a  =  f  W*r  cos  a,  therefore  during 
the  powder  period,  B1=K-Wrsin8J-f  [  Pbsin(0+a)+Wrcosa] 

in  the  recoil,  B=K-Wrsin0-fWrcos  a  ,  and  the  charge 
of  required  braking,  becomes,  B-Bt»  fPbsin(0+a) 

BEACTION3  ON  THE  BOTTOM  CARRIAGE. 

The  reactions  on  the  bottom  carnage  are:- 

(1)  Qt  and  Q2  reversed,  the  roller 
reactions  on  the  chassis  of  the 
top  carriage  (Ibs) 

(2)  V  reversed,  the  braking  reaction 

(Ibs) 

« t- 

(3)  The  horizontal  pintle  bearing 

reaction  n. 

(4)  The  weight  of  the  bottom  carriage 

Wtc- 

(5)  The  traversing  roller  and  clip 
reactions  Rt  and  R8  (Ibs) 

Then,  resolving  forces  along  and  normal  to  the 
chassis,  we  have, 

(Qt+Qa)sin  u  +B-HCOS  a+ (Rft-Rt )sin  a-Wtcsin  a  =0   (I1) 

(Q  +Q  )cos  u+Wt.cos  a-Hsin  a  -(R  -R  )cos  a  =0    (21) 
a   t          wO  a   i 

and  for  moments  about  the  trunnion, 

H  dh-Bd^Q^-Q^-R^-R^-W,^  =0  (3  •  ) 

where  x^  =  the  momentum  of  Wtc  about  the  trunnion. 

BXTSRNAL  REACTIONS  ON  THE  SYSTEM  CONSIST- 
I  NG  OP1  THS  TOTAL  MOUNT. 

Adding  equations  (1)  and  (I1),  we  have, 

dal 
Fbcos(0+a)-Wrsin  a-mr-— •  -Hcos  a+(Ra~Rt)sin  a-Wtc 

sin  a  «  0  (1") 


901 


Since  Pbcos(0+a)-mr  —  pK  and  »r+Wtc=Ws  the  total 
d  t 

weight  of  the  mount.  Equation  (I11)  reduces  to, 

K-Hssin  a  -  H  cos  a  +(Ra-Rt)sin  a  =  0  (1") 

Adding  (2)  and  (2'),  we  have 

Pbsin(0+a)+Hrcos  a+Wtccos  a-Hsin  a  -(R2-R1)cos  a  »0 

(2") 
Adding  (3)  and  (31),  we  have, 


-  (y0cos  a+x0sina)»0  (3B) 
dt* 

Eliminating  (Ra~Rt)  from  (1")  and  (2"),  we  have 

Kcos0-H+Pbsin  a  sin(0+a)  =0  (a) 

Eliminating  H  from  (1")  and  (2"), 

(R2-Rt)+Ksin  a-Pbsin(0+a)cos  a-Ws=0  (b) 

and  equation  (3")  reduces  to  for  moments  about  the 
trunnion, 


sina)=0  (c) 

From  (a)  and  (b),  we  have,  H=Kcos0+Pbsina  sin(0+a) 
R>-R1»Pbsin(!^+a)cos  a+»s-K  sin  a.   Substituting  the 

value  of  H  in  (c)  and  combining  with  (b)  we  obtain 
R(  and  Rt  respectively. 

PERCUSSION  REACTIONS: 


The  percussion  reactions  take  place  during  the 
powder  period  and  are  reactions  of  a  magnitude 
comparable  with  the  powder  forces.   In  an  ordinary 
cradle  recoil,  the  direct  effect  of  the  powder  re- 
actions are  practically  eliminated  by  allowing 
the  gun  to  recoil  along  the  bore.   In  mounts  of 
the  chassis  type,  especially  when  the  gun  elevates, 
we  have  a  large  component  of  the  powder  reaction, 
which  causes  the  chassis  to  offer  a  corresponding 
reaction. 


902 


OA/ 


Fig. 7 


903 


FERCUSStQH  REACT/ON 
ON  RECO/L/N6  PARTS 


PERCUSSION  REACT/ ON  ON  TOP  CARRIAGE 


*J 


t  \ 


Fig.  8 


904 


In  dealing  with  impulsive  forces,  the  effect 

of  continuous  or  finite  forces  is  negligible  com- 
pared with  the  percussion  reactions. 

Hence  in  the  following  we  will  omit  such  forces 
as  gravity,  and  the  recoil  brake  reaction. 

PERCUSSION  REACTIONS  ON  THE  RECOILIN3  PAHTS: 

The  percussion  reactions  are, 

(1)  The  powder  force  —  Pb 

dal 

(2)  The  inertia   resistance  I=ffi_—  —  • 

dt2 

(3)  The  resultant  reaction  of  the 
chassis  ---  Q 

P^  acts  along  the  bore,  I  acts  parallel  to  the 

chassis  and  through  the  center  of  gravity  of  the 
recoiling  parts,  while  Q.  balances  these  reactions 

at  their  intersection,  as  shown  in  fig.(8)« 
The  force  polygon  of  the  percussions  is  abc, 
where  a  b  //P^,  bc//I  and  ca//Q.   The  direction  of 
Q.  is  slightly  inclined  to  the  chassis  due  to  the 
friction  angle  u.   Further  Q  is  the  resultant  of 
Q  and  0  the  front  and  rear  roller  reactions.   Now 

t          2 

the  resultant  of  Q  and  I,  must  intersect  the 
resultant  of  P^  and  Q8  .  Since  P^  intersects  at 
02  at  a,  we  have  the  direction  of  the  resultant 
of  Q,  and  I  along  ae  .   In  the  force  polygon  bd  is 
drawn  parallel  to  ae,  and  therefore  cd  is  proportion- 
al to  Ql  while  da  is  proportional  to  Qa  . 

In  the  force  polygon,  we  have, 

Pb   I    Qt   Q2  vc  . 

—  =  —  =  —  =  —      hence  I  =  —  P^, 

ab   be   cd   da  ab 

""^* 


DYNAMICAL  RELATIONS  ON  FIRING     Small  guns  up  to 
FROM  AN  AEROPLANE.  a  caliber  of  75 

have  been  successfully  fired  from  large  aeroplanes, 


8/ 


905 


Larger  calibers  may  be  possible  by  the  introduction 
of  the  muzzle  brake,  which  thereby  reduces  the  re- 
coil reaction. 

In  this  discussion,  however,  we  will  take  the 
simple  case  of  a  gun  without  a  muzzle  brake.  Let 
VQ  =  horizontal  velocity  of  the  plane  before 

firing  (ft/sec)  Vo 

V  a  velocity  of  the  plane  immediately  after 
firing  (ft/sec)  Vt 

Vr  =  velocity  of  the  gun  at  the  end  of  the 

powder  period  (ft/sec)  V^ 
v  =  muzzle  velocity  of  projectile  (absolute) 

(ft/sec) 
Pb  -  powder  reaction  (Ibs) 

R  =  recoil  reaction  (Ibs) 

inr  and  wr  =  mass  and  weight  of  recoiling 

parts  (Ibs) 
ng  and  *s  ~  mass  and  weight  of  equivalent 

weight  of  aeroplane  +  weight  of 
cradle  and  mount  (Ibs) 

Assume  the  gun  to  be  fired  horizontally  while  the 
aeroplane  flies  horizontally: 

During  the  powder  period,  we  have  the  mutual 
impulsive  reaction  between  the  gun  and  aeroplane  = 
Fb  dt  t 

For  the  gun,  /  *  Pb  dt  =  mr(Vo-Vr)          (1) 
o 

the  impulsive  effect  of  the  recoil  reaction  R  being 
negligible.   For  the  projectile  and  powder,  we 
have,  Ato 

f    Pb  dt  =(m+0.5m)(v-V0)  during  the  travel 

up  the  bore. 

N  7~Vo 

Pb  dt  =  mU700-(— )  during  the  powder 

°  expansion. 


906 


At 

/   Pb  dt  *  I(v-V0)+i4700  *  mv+m4700  (approx) 

0  (2) 

Let  us  DON  consider  the  effect  of  the  recoil 
reaction  R  on  the  aeroplane  and  fixed  part  of  the 
mount.   On  firing  the  aeroplane  the  aeroplane  acts 
somewhat  as  an  elastic  beam,  more  or  less  supported 
by  the  air  reactions  at  the  ends.   We  may  consider, 
the  equivalent  mass  of  the  aeroplane  and  mount  at- 
tached =  0.7  to  0.8  the  actual  mass  of  the  plane 
and  mount.   We  will  denote  ms  as  this  equivalent 
mass. 

Then,  for  the  motion  of  the  aeroplane  during 

the  recoil  period,  we  have, 
«S(V0-V  ) 

R  =   S   °   f     Ubs)  (3) 

t 

and  for  the  motion  of  the  recoiling  parts  during 
this  same  period, 

nr(V  -Vr) 

R  =     *       (IDs)  (4) 

t 

since  the  recoiling  parts  must  have  the  same 
velocity  as  the  plane  at  the  end  of  recoil. 

It  is  interesting  to  note  the  magnitude  of  the 
relation  of  the  various  velocities  for  a  typical 
small  mount. 

V0  =  100  miles/hour  =  146.6  ft/sec. 

V0-Vr»  30  ft/sec,   roughly;  Vr  =  116  ft. sec. roughly, 

V  =  between  116  and  146  ft/sec,  say   130  ft/sec. 
Thus  vie  have  a  check  in  the  velocity  of  the  plane 
of  several  feet  per  second,  the  magnitude  of  which 
depends  of  course  on  the  ballistics  and  relations 
of  the  various  masses. 

Combining  the  previous  equations,  we  have, 

•r(V0-Vr)=mv+i  4700  (5)  ' 


(Wg+«r)(V0-Vt)«mv  +1  4700  (7) 

That  is,  as  we  should  expect  from  first  principles, 


907 


the  momentum  imparted  to  the  aeroplane  backwards, 
equals  the  momentum  imparted  to  the  projectile 
and  powder  forwards. 

Let  us  now  assume  the  recoil  reaction  con- 
stant, and  let  b  equal  the  length  of  recoil. 

Now  due  to  the  superior  motion  of  the  aero- 
plane as  compared  with  that  of  the  gun,  during 
the  recoil  the  aeroplane  does  work  on  the  gun,  in 
bringing  the  velocity  from  the  smaller  value  Vr  to 
the  larger  value  V  hence 


2 
The  energy  taken  from  the  aeroplane,  becomes 


2 

hence  -R  b  =  -|  (V*-V»)  -  -|(V0-?«)   therefore, 

the  recoil 

reaction,  becomes        _  v«     »8     + 

_  lr  ^"o   ""rvr.  xms*mr.ri, 

R%-[  (-r  *  — >-(-T— )v; ' 

(Ibs) 
.hare  Vr=Vo  -(211i|225,  ({t/sec) 

aod  vvo-(HIiZ22i)  ((t/seo) 
ms+mr 


DISAPPBARISG  AHP  OTHBH  TYPgS  OP  CARBIA6E8. 

TYPES  OP  DISAPPEARING      Disappearing  gun  car- 
CAFRIAGES.  riages,  as  evident  by  their 

terminology,  are  designed, 

so  that  in  the  recoil  the  gun  is  brought  down  below 
a  parapet  and  disappears  from  the  enemy's  view. 

The  gun  is  loaded  in  the  lower  position.  By  in- 
troducing a  counterweight,  the  gun  is  brought  by 

gravity  to  the  firing  position,  the  gun  during 
the  firing  period  only  being  above  the  parapet. 
Disappearing  gun  carriages  may  be  broadly 


908 


classified  in  two  general  types:- 

(1)  Revolving  or  rotating  types, 
where  the  gun  lever  rotates  about 
a  fixed  axis,  as  in  the  Monorieff. 
Howell  and  Krupp  carriages. 

(2)  Sliding  carriage  types,  where 
the  Cardon  system  of  linkage  is 
used,  the  gun  lever  being  constrained 
to  move  at  two  of  its  points  along 
guides  practically  at  right  angles, 
as  in  the  Buffington  Crozier  models. 

APPROXIMATE  THEORY  OF  THE     The  following  as- 
ROTATING  TYPE  OF  DISAPPEAR-  sumptions  are  made  and 
ING  CARRIAGE.  the  validity  of  these 

assumptions  will  be 
considered  more  in  detail  later:- 

(1)  The  center  of  gravity  of  the  gun 
will  be  assumed  at  the  gun  trunnioni 

(2)  The  angular  displacement  of  the 
gun  lever,  during  the  powder  period, 
will  be  assumed  small  and  will 

therefore  not  effect  the  initial 
geometrical  conditions  greatly. 

(3)  The  inertia  effect  of  the  elevat- 
ing rods,  will  be  assumed  negligible 
as  compared  with  that  of  the  gun, 
lever,  gun  and  counterweight. 

(4)  The  elevating  arm,  will  be  assumed 
approximately  parallel  to  the  axis 
of  the  gun  lever  and  roughly  equal 
to  the  upper  half  of  the  gun  lever. 

(5)  The  angular  movement  of  the  gun 
itself  during  the  powder  period  will 
be  assumed  very  small. 

From  assumptions  (3),  (4)  and  (5)  we  may 
neglect  the  reaction  of  the  elevating  arm  during 
the  powder  action  period,  for  the  following  reasons: 
(a)     The  tangential  component  of 


909 


the  elevating  arm  reaction 
becomes  zero  due  to  assumption 
(3). 

(b)     Condition  (4)  assumes  the 
instantaneous  center  of  the 
gun  practically  at  infinity. 
Hence  the  angular  velocity  of 
the  gun  at  the  end  of  the 
powder  period  is  negligible; 
the  angular  acceleration 
therefore  may  be  assumed  zero, 
and  the  normal  reaction  of  the 
elevating  arm  becomes  zero. 

In  practice  it  is  possible  to  obtain  (1)  com- 
pletely, and  (2)  and  (3)  are  closely  realized.  The 
condition  (4)  may  be  met  constructively  at  one 
elevation  but  is  difficult  to  meet  for  all  elevations, 
since  the  gun  customarily  is  designed  to  recoil  to 
the  same  loading  angle. 

To  reduce  the  reaction  on  the  elevating  arm 
it  is  customary  to  introduce  a  kick  down  buffer  at 
the  bottom  end  of  the  arm,  and  thus  during  the 
powder  period  a  small  minor  reaction  comparable  with 
the  buffer  resistance  is  introduced  between  the 
elevating  arm  and  gun.  This  reaction  may  be  neglected 
as  compared  with  the  major  reactions  of  the  gun 
lever. 

Therefore,  as  a  first  approximation,  however, 
we  will  neglect  the  reaction  of  the  elevating  arm, 
and  assume  the  center  of  gravity  of  the  gun  located 
at  the  trunnions.  Let 

Wg=  weight  of  the  gun  (Ibs) 

Wr=  weight  of  the  gun  lever  (Ibs) 

wcw=  weight  of  the  counterweight  (Ibs) 

Ir=WrkJ  =  moment  of  inertia  of  gun  lever 

about  fixed  axis  of  rotation. 
Icw  =  Wcwk§w  =  moment  of  inertia  of  counter- 
weight about  fixed  axis  of 
rotation. 


910 


REACTIONS   ON  THE.    ROCKEIR  AT    GUN 
DURING   POWDER    PERIOD 


REACTIONS  ON  THE    ROCKER    AT  GUN 
AFTER    POWDER    PERIOD 


w 


cw 


Fig.  9 


911 


T  and  N  =  tangential  and  normal  trunnion  re- 

action (Ibs) 

X  and  Y  =  horizontal  and  vertical  reactions 

at  axis  of  rotation  of  gun  lever   (Ibs) 

P  *  total  powder  reaction  (Ibs) 

Pffl  =  maximum  powder  reaction  (Ibs) 

0  =  angle  of  elevation  of  gun 
6^=  initial  angle  of  gun  lever  with  respect 

vertical 
0£=  final  angle  of  gun  lever  with  respect  vertical 

r  =  radius  of  upper  half  of  gun  lever       (ft) 
r1  =  radius  to  center  of  gravity  of  counter- 

weight (ft) 

R  =  reaction  of  oscillating  cylinder  brake 
d^=  initial  angle  R  makes  with  the  normal  to  r' 
ro  =  distance  from  axis  along  r1  to  line  of 

action  of  R. 
m  =  mass  of  projectile 
IE  =  mass  of  powder  charge 

v  =  muzzle  velocity  (ft/sec) 

TfVf  =  total  friction  torque  resisting  rotation 

From  fig.(  9).   the  gun  axis  makes  an  agle  0-  9j 
with  the  tangent  of  the  path  in  the  initial  position 
of  the  gun. 

For  the  motion  of  tte  gun  lever,  we  have  for 
moments  about  the  fixed  axis, 


+*  cos  d  •  ro  +  Vf  (1) 

and  for  the  motion  of  t"he  gun  along  the  tangent  to 
its  initial  path,  „ 

Pcos(?-  6i)-T-  -*  r  —         (2) 
g   dt» 

If  s  =  the  displacement  along  the  arc  of  the  gun 

trunnion 

V  =  the  corresponding  tangential  velocity  of 
the  gun  trunnions, 


912 


ds      d6    d*s   dV     d*e 

—  *  r  — ;          •  =  r  -      Hence,  combining 

dt      dt    dta   dt     dt*   the  two  equations, 

we  have, 

wg  *r  *cw  dV       ro    rf 

Pcos(0-6.j  )=( — +— -+- r—)— +Rcosd;  — +Tf  —        (3) 
1    g   r2  ra   dt        r    •  r 

Evidently  — J+~"£"~  may  be  regarded  as  the  so  called 

equivalent  translatory  mass  at  the 

gun  trunnions  due  to  the  rotational  inertia  effect 
of  the  gun  lever  and  counterweight. 

Integrating  equation  (3),  we  have, 

r0  rf 

Rcosd  —  dt  Tf—  dt 

Pcos(0-6i)              r  c  r 

V  =  / — — dt  -  /  -— / 


g  Ir  Icw         Wg  Ir  Icw       Wg  Ir  Icw 

^P   I          -f-_^__  "   I    HM   I  ----  -,_S   \  \   -- 

g   r2   r2          g   r2   r2       g   r*   r2 

Now  both  0  and  d  as  well  as  the  friction  torque 
TfTf  vary  during  the  powder  period  but  as  the 
change  is  small,  we  are  quite  justified  in  assum- 
ing them  constant.   Further,  since,  Pdt=(m+0.5m)v 
(during  the  travel  of  the  shot  up  the  bore),  we 
have 


(m+0.5m)v 
—  -  -   dt   =  -  -  —  -  -  cos  (0-°i)    or   in   terms 

Wrf   1  r   i  /%  n-  W/<   1—   -l-riui 

8  ,  r  ,  cw        (_5+_j,  cw\  of  the  free 

g   r2   ra         g   r2   r2  velocity  of 

recoil,  (m+0.5m)v 

Vfcos(0-e.)=  -  -  —  -  -  cos  (f?-9i) 

W 

wr  xr  1cw. 

(  —  +  —  +  -  ) 

g   r2   r2' 

where  Vj  is  the  equivalent  free  velocity  with  a 

recoiling  mass  equal  to 

"r  lr  xcws 

(  —  +  —  +  -  ) 

g   r2   r2' 

Integrating  again,  we  have 

Xfeos(0-ei)=["*0'5mi  —  ]x'  008(0-6^  where  x1  =  the 

g  ^  r  |  cw  absolute  dis- 

&   r  placement  of 
the  projectile  up  the  bore.  Now 


913 


•-*) 
x'cos(0-6i)=  u  cosdy-QiJ-XfCosCCf-Si)  hence,  we  have 


wg  lr  Jcw 

—  +  —  +  -  +m+0.5m 

g   r2   r2 


Ir  Icw 


now  m+0.5m  is  small  compared  with  —  +-T+-—  r  bence 

g   r2  r2 

we  may  assume 

(m+0.5i)ucos(0-ei) 
Xfcos(gf-6i)  =  -  -  —  -  -  i  —    (ft) 


The  equations  of  recoil,  become  therefore 

^2+Tf  —  )t 
r   rr 


V=VfCos((?-9i) 


and 


Ic, 

'"r1 


(Rcosd— +Tf — )t* 
r   Ar 

(approx) 


(m+0.5m)v            (m+0.5m)  u 
where  Vf=  and  X         


wr  ^  Jcw  ,       ,wg  Jr  ^w. 
( — +—+ — r)       ( — +-T+ — r)  +  m+  0.5  m 
g   r2   r2          g  r2   r2 

We  see  the  equations  of  recoil  during  the 
powder  period  are  exactly  similar  to  the  previous 
recoil  equation,  the  recoiling  mass  now  including 
the  inertia  effect  of  the  rotating  elements.   Hence 
the  previous  interior  ballistic  formulas  are  im- 
mediately applicable  for  the  computation  of  the 
free  recoil  displacement  E  and  the  time  of  the 
powder  period  te. 

For  the  maximum  velocity  of  recoil,  we  have 

mv +47001 
Vfm  =  "    — (ft/sec)   and  the  max. 

^-£+:£?L  velocity  of 

&  r   r  constrained 

recoil  along  the  path  of  the  gun  trunnion,  becomes, 


914 


(Rcosd— +Tf—)tc 
r     r 


IS+T  — 
r     r 


The  corresponding  maximum  angular  velocities 
and  angular  displacements,  become, 

,de.  V"          s. 

"«*(dI)m=T   and   e*  "  T 

The  energy  of  recoil  at  the  end  of  the  powder 
period  becomes,  w 

A  s  i(T  +T   +  — i  r9 )n9 
Hm   a v  r  xcw  TT     m 

From  the  energy  equation  we  may  easily  consider 
the  remainder  of  the  recoil. 

Since  the  brake  and  friction  resistances  are 
small  compared  with  the  powder  reaction  and 
the  inertia  resistance  of  the  rotating  parts,  we 
may  assume  with  sufficient  accuracy  that 

vm=vfmcos^0~9i^  and  sm  3  B  cos(*-0i)   We  have, 
for  the  recoil  energy  at  any  angular  displacement 
6. 


f     (Rcos  d.r0)d9  +  /  '    Tf rfde+Wcwr ' (cos  Qj-  cos  6) 
0i  9i 

-  Wgr  r(cos  Qj  -  cos  9)=  Am-  A   where  Wgr  =  weight 

of  gun  and  that 

portion  of  the  rocker,  not  including  the  counter 
weight  reduced  to  an  equivalent  weight  at  the  gun 
trunnion,  that  is 


T"  r"  =  u  Wg  where  r'  =  radius  of 


915 


Wrrr 
*gr=Wg+  '         rr  =  distance 

from  axis  to 

center  of  gravity  of  rocker.   Since  d  varies  with 
the  angular  displacement  of  the  gun  lever,  from  a 

layout  we  may  readily  evaluate  the  term 

8 
f   (Rccs  d  rQ)de  provided  R  is  assumed  constant 

i  which  is  usually  the  case. 

Further  since  T^r£  does  not  vary  greatly  we  may 
assume  it  constant.   As  a  close  approximation, 

u(Wg+Wr+Wcw)rf*     u  =0.15  roughly 

radii 

bearing  of 
axis  of  rotation  of  rocker 

r"  =  radius  of  trunnion.   Further  T*rf=T  Irf+Tflr" 
f  f  f  f  f 

Hence    9* 

/    Tfrf  de  =  Tfrf  (6f-  6i)   (ft/lbs) 

now  A  =  r(Ir  +  Icw+  —  ra)ws 
2 

(rad/sec) 
REACTIONS  OK  THE  CORDAN  LINKAGE     Reactions  on  the 

DISAPPEARING  CARRIAGE  DURING     Gun:    The  center 

THE  POWDER  PERIOD.  of  gravity  of  the 

gun  is  assumed  at 
the  trunnion  axis 

of  the  gun.   The  angular  acceleration  of  the  gun 
is  assumed  small  and  the  reaction  of  the  elevating 
arm  on  the  gun  is  considered  a  secondary  force,  this 
being  possible  by  a  proper  arrangement  of  the  parts 
or  by  the  introduction  of  a  kick  down  buffer  at  the 
base  of  the  elevating  arm. 

The  primary  reactions  on  the  gun  consist: 

(1)     The  powder  force  along  the  axis 
of  the  bore  =  Pb 


916 


f?£ACT/OA/S  ON  T/if  GUN 


. 


Fig.  10 


917 


(2)  The  trunnion  reactions  divided 
into  horizontal  and  vertical  com- 
ponents X  and  Y  respectively. 

(3)  The  weight  of  the  gun  acting 
through  the  trunnion  axis  =  Wg 

(4)  The  tangential  inertia  force 
along  the  path  of  the  movement 
of  the  trunnions  or  normal  to  a 
line  from  instantaneous  axis  to 

the  trunnion  axis  = 

d2s 

•  nirf 

*dt* 

(5)  The  centrifugal  inertia  force, 
normal  to  the  path  of  the  trunnion 
axis  and  proportional  to  the  square 
of  the  angular  velocity  =   dQ 

sftp* 

The  secondary  reactions  on  the  gun  are: 

(1)  The  elevation  arm  reaction  on  the 
gun  comparable  with  the  kick  down 
buffer  reaction  at  the  base  of  the 
elevating  arm. 

(2)  The  inertia  couple  due  to  the 
angular  acceleration  of  the  gun 
about  the  trunnion  axis.  . 

In  the  following  analysis,  we  will  neglect 
the  effect  of  the  secondary  reactions.   The  forces 
on  the  gun  neglecting  the  secondary  forces  are 
shown  in  fig . ( 

Since  we  assume  the  rotation  negligible,  we 
have  the  equations  of  motion, 

Pbcos  0-mg  —  cos  B+mgl(-— -)2  sinB-X=0 

T-mg^|  sinB-mgk(~)2  cos  B+Wg-Y  =0 


b  d2s   d*6 

where  tan  B  =  — -  tan  6   ;  — -=  1— -r  approx.  since 
a+b          dt1*   dtz 

1  does  not 

change  greatly  during  the  powder  period. 


918 


ON  TH£   GUN 


FACTIONS  ON  TflF  SUD//V6 


Fig.  I 


919 


I  =  /  (a2  +  2ab)cos26+ba   Hence,  the  trunnion  re- 
actions become, 

d28         d6 
IT— rcosB+m,*  1  (~)2  sinB 

at      s  at 


Y  -  Pvsinef-D-1—  —sin8-mcl(-—  -)2cosB+Wtf 
5  dt       *  dt 

BUCTIOMS  OK  THE  BOCKKB. 

The  reactions  on  the  rocker,  are: 

(1)  The  reaction  of  the  gun  on  the 
rocker  divided  into  components  X  and 
Y. 

(2)  The  reaction  of  tlie  sliding  car- 

riage on  the  rocker  at  the  rocker 
trunnion,  divided  into  components 

X1  and  Y1. 

(3)  The  reaction  of  the  counterweight 
cross  head  at  the  wrist  pin  of  ttie 
cross  head,  divided  into  components 
X"  and  Y". 

(4)  The  weight  of  the  rocker  at  the 
center  of  gravity  assumed  at  the 
rocker  trunnion  Wr. 

(5)  The  rotational  inertia  couple 
due  to  the  angular  acceleration  of 
the  rocker  =   ,2 

Ird~t7 

(6)  The  tangential  inertia  force  of 
the  rocker  along 

d*x 
OX  *  mrI7jr  actinfi 

through 

center  of  gravity  of  rocker. 
(7)     Tbe  centrifugal  inertia  force 
of  the  rocker  normal  to 
OX 


at 

The  equations  of  motion  of  tte  rocker,  become, 
along  OX  - 

Y-Yi+Y"-m  —  —  =Q 

x  *  A  "     ° 


920 

~fri 

along  OY 


about  the  instantaneous  axis  I, 

X(a+b)cos  9+Y  b  sin  0-X'a  cos  9-Y"  a  sin  9  -  mr 

dfx  d29 

_  a  cos  e  -  lr  _=  o 

BBACTIOHS  01  THB  SLIDIHG  CARRIAGE  AHD 
COOMTBR  WEIGHT  RB3PBCTIVSLY  IN 

THB  DIRECTION  OF  THBIB  MOTIONS. 

Considering  the  sliding  carriage,  we  have, 

d*x 

X1— H-mc— — -=0   Where  R  is  the  hydraulic  brake  re- 
action en  the  carriage  and  for  the 

counterweight, 

d  y 

Y"-mcw w  =0 

dt2 

EQUATION  OF  MOTION  OF  THE  SYSTEM  DURING 
THE  POWDER  PRESSURE  PERIOD. 

Substituting  Vhe  values  of  X',Y",X  and  Y  in 
the  moment  equation  about  the  instaneous  axis  of 
the  rocker,  we  havs, 
Pb  I  (  a  +  b  ) 

cos  e  cos  0+bsin  9]-m,.  [  (a+b  )2cosa6+b2sin29] 

*  dt2 


+Wgb  sin  9-[R*(mr+mc) ]a  cos  9  mcw  — —•  a  sin  9 

*  dt2  dt 

-Wcw  a  sin  9  -   lf  1J[  =  0 

Now  x  =  a  sin  9 

dx  d6 

—  =  a  cos  9  — 
dt          dt 

d2x  d29          He,, 

jpj-  =  a  cos  9  — —  a  sin  9  (-^)2 


921 


and  y  *  a  cos  9 

dy  d9 

—  ~  -  a  sin  9  — 
dt  dt 

d*y  d*9  de,9 

_  =  -  a  Sln  0  _  -  a  cos  6    (_)• 

If   we   assume   the  positive  direction  of   y   upward, 
then 


Substituting  these  values  in  the  arova 
equation  we  have  the  general  dynamical  equation 
of  the  disappearing  carriage  during  the  powder 
pressure  in  terms  of  a  single  coordinate  variable 

e. 

The  differential  equation  of  motion,    becomes, 
Pb[(a+b)cos   9   cos(8+b   sin  6   sin  0]-nig[  (a+b)*cos*  9+t>» 

j  2  ft  j  j  A 

sin8   9]   -  •'•W.jb   sin  9-B   a  cos  9-(mr+mc)a2cos*  9  - 
dt*  dt* 

d29  do 

-m_wa2    sin2   6   —  -     +(m..-»-m..)a  sin  9   cos   9(  —  )*   -mrw 
dt2  dt 

a  sin  9   cos   9(~)«  -Wcw   a  sin  9  -  I     —  *0 
dt  dt* 

Combining   terras,    we    have,    Pb[(a+b)cos  9   cos0+b   sin   9 

sin0]-   <ing    t(a+b)2   cos2    9+b2    sin2   9]4(mP+mc)a2   cos2 

1    d29 
e+mcwa2    sin2  9  +  1  I+t  (mc^pJa  sin  9  cos  9   -  mcw 


d0 
a  sin  9  cos  9]  ( — )a   -R  a  cos  9+W,,   b   sin  9  -Wcwa  sin 

dt 

0=0 

The  equation  is  in  the  form  of  A— — +B( — )2+C=0 

dt2   dt 

where 


922 


A=«g[(a+b)acosa   6+b«sina 


sin  6    +Ir 


B=   -[(mc+nr)a   sin  9   cos  6   ™Cwa   s*n   9   cos   ®^ 

C=  -Pb[  (a+b)cos  6   cos  tf+b   sin  8   sin  0]+Ra  cos  6-Wg 


b   sin  6    +WCW   a  sin  6 


CALCULATIOM  Of  THE  RECOIL  DUBIHG  THE 
POWDBR  PRESSURE  PERIOD. 

The   general   equation  of   motion   for   the   system, 
becomes, 

pb[(a+b)cos   6   cos0+b    sin0  sin0]-[mg(a+b  )*   cos*   8 
+b2    sin2    6]  +S;TDr+mc)a2   cos8   6    +mcw   a2    sin8    8+Ir] 


— -   +[(mc+nr)a   sin   6   cos  8  -mcvf   a   sin  8   cos   8]  (— 

Q   V  d  tr 


-R  a  cos  e+*gb    sin   8-Wcw   a  sin   8=0 
We   nay  urite    this,    as 


APv-B  -  +C(  —  )2-D  =0     where   A  =(a+b  )cos8cos0+bsin8sinCf 
dt2        dt 

B  »e»g[  (a+b)acosa8+hasin8] 
+  [  (mr+nic  )  a2cos8+mc|f  a2 


C  =    (mc-*-»r)a   sin6cos6-mC)|a  sin8cos8 
D  =Ra  cosSHfb   sin  e  +  Wa  sin  8 


Integrating,  we  have 

t  J/\  t 


now  f  Pjjdt  =(n+0.5l)v  during  the  travel  up  the  bore 

where  m  =  mass  of  the  projectile 

a  =  mass  of  the  powder  charge 

v  =  velocity  of  the  projectile  in  the  bore. 
hence 


923 


OF 


y     FOX 


*  INSTANTANEOUS  C£NT£#  OF  &OTAT/ON 


.    12 


924 


-7  s  |(m+0.5m)v+  -  J    (-7)*  dt  -  -  t  which  is  the 

at     o  D  Q      ut    V\\    B 

general  ex- 
pression of  the  angular  velocity  of  the  system  during 

the  travel  up  the  bore.  Integrating  again, 

6=  -(m+0.5m)u+-  /  /   (-7)*  dt.dt  -  •£-  t*  where  u» 

0  D   A    n      (It  GO 

the  travel 
up  the  bore 

These  equations  may  be  integrated  by  a  point  by  point 

method . 

KINEMATICS  OF  A  CORDAN  LINKAGE  In  the  analysis 
DISAPPEARING  CARRIAGE  INCLUDING  of  the  kinematics 
EFFECT  OF  ELEVATING  ARM.  of  any  linkage, 

we  have  two  sys- 
tems of  diagrams, 

viz:-  velocity  diagrams  and  acceleration  diagrams. 
By  the  use  of  the  velocity  diagram,  we  may  calculate 
the  centripetal  accelerations,  which  of  course  must 
be  included  in  the  acceleration  diagrams.   Due  to 
the  required  velocity  diagram,  we  are  justified 
in  using  the  instantaneous  center  about  which  the 
gun  lever  rotates. 

Therefore,  in  considering  the  instantaneous 
center,  the  cordan  linkage  including  the  elevating 
arm,  becomes,  a  four  bar  linkage  and  we  may  con- 
struct a  velocity  diagram  as  for  any  four  bar 

linkage.   The  acceleration  diagram  of  the  linkage, 
however,  is  not  theoretically  equivalent  to  a  four 
bar  linkage,  since  the  instantaneous  center  of 
the  gun  lever,  has  a  definite  path  in  the  recoil. 
Hence  the  distance  from  the  Instantaneous  center 
of  the  gun  trunnion,  changes  in  ths  recoil  and  the 
tangential  acceleration  becomes, 

d (wl)     dlA  ,  dw 

•  *»  -  «•  1  -    Since  1  does  not  charge  great- 

dt  dt   ly  during  the  powder  period 

we  are  justified  in  omitting 


925 


dl  d(wl)    dw    d«9 

w  —-  being  small  then  — —  *  1—  i  — 

dt  dt     dt    dt* 

Consider  any  position  of  the  linkage  during  the 

powder  period:   see  fig. 03). 
Let 

9  *  angle  made  by  gun  lever  or  rocker  with 

vertical  (rad) 
1  =  distance  from  instantaneous  center  to 

gun  trunnion  (ft) 

B  =  angle  made  by  1  with  vertical  (rad) 
d  =  distance  from  gun  trunnion  to  elevating 

arm  trunnion  on  gun  (ft) 
0  =  angle  d  or  axis  of  bore  makes  with 

horizontal 

C  =  length  of  elevating  arm 
Q.  =  angle  made  by  c  with  vertical  (rad) 
x0  and  y0  =  coordinates  of  base  of  elevat- 
ing arm  (ft) 

dfl 

w  =  —  =  angular  velocity  of  gun  lever  (rad/sec) 
dt 

dQ 
wi  -  —  =  angular  velocity  of  elevating  arm 


VELOCITY  DIAGRAM. 


The  linear  velocity  of  point  0,  becomes, 


dt 

The  component  along  "d"  becomes,  Iw  cos(#-B) 
Tbe  linear  velocity  of  point  Q1,  becomes,  cw1 

Its  component  along  d,  becomes,  cw1  cos(0-ft) 

Hence  Iw  cos(0-B)=  cw '  cos(0-a) 

1  eos(0-B)  dO  1  cos(0-B)  de 
and  "'  ''  c  cos(0-Q)  "  '"•  dt  "  c  cos(0-Q)  dt 
The  angular  rotation  about  the  trunnions,  equals, 


926 


FOX    PO/NT  O, 


.  wc- 
oo, 


13 


827 


d0  _   Vel.O-Vel.O      _   H'C   sin(0-Q)-wl   sin(0-B) 
dt  "  d  d 

...    cos  (£)-B)  _,  rt.  ,,.,  Ov,w 

-Q)-l   sin(0-B)]- 


(rt  *\ 
cos(0-Q)  d 

7   [tan(0-Q)cos(0-B)-sin(0-B)](4^)  (2) 

d  dt 


ACCELERATION     DIAGRAM: 


The  acceleration  of  point  0,  the  center  of 
gravity  of  the  gun,  along  the  x  axis, 

d*x       d*8         ,d8, 

_  g  ,  1  —  cos  B  -  1<-)2  sin  B 

Along  the  y  axis, 


dtz   dt2       dt 

Tbese  values  do  not  include  the  effect  of  the  small 
change  in  "1"  in  the  powder  period.   To  include  this, 
we  have  Xg=(a+b)sin  9 

dxg  d9 

—  —  =(a+b)cos  8  — 

dt  dt 


=(a+b)  cos  8   -  -(a+b)sin  e*    (3) 
dt  dt 


and  y^=  -  b  cos  Q 

dyg        de 

-  =  b  sin  9  — 
dt  dt 

j  r 


=  b  sin  e+b  C08  e  ()»         (4) 

dt*          dt8         dt 
The  acceleration  of  0',  is  divided  into  the  follow- 

ing components: 

(1)     The  acceleration  of  0,  divided 
into  two  components 
daxg     d«vg 
dt2      dt2 


928 


(2)  The  centripetal  acceleration 
of  O1,  about  0  directed  along  d 
towards  0  * 


dt 


(3)     The  tangential  acceleration  of 
0',  about  0,  normal  to  d  and  equal 


dta 

Since  0',  is  a  common  point  for  both  the  gun  and 
the  elevating  arm,  we  have,  also,  the  acceleration 

of  O1,  divided  into, 

(1)     The  tangential  acceleration  of 

the  gun  lever  at  0 
d«Q     dw' 


(2)     The  centripetal  acceleration  of 
the  gun  lever  at  0 

-  c(—  )*  *  c*1* 
dt 

"From  the  acceleration  diagram,  we  have  the 
following  vector  equation 


dt«  dt»    dt    dta  dt*   dt 

The  two  unknowns  in  the  equation,  are 


d*Q 
^—  and  c  -  which  we  will  denote  by  ad  and  ac 

dt»       dta 

bat  we  have  two  coordinate  equations 

along  ox  and  oy  and  hence  a  solution  is  possible. 

The  solution  may  either  graphical  or  analytical, 
He  have,  from  (3)  and  (4) 


.(a.b)t=o,9     -  sin  e() 
at»  dt1        dt 


929 


!•(-> 

d  (—)**[  tan  (0-Q  )  co  s(0-B)-sin  (0-8)]*   -  —  — 
•It  i 

,dQ   »  ,«        1«    cos«(0-B)          d6.» 

c(  —  )      -cw1      =   -     -      (  -  )        From   the   ac- 

dt  C       COS8(0-Q)  dt  celeration 

fU    J.C    e*    fce*»?    rj**,",*     -rj  ••fc?'?    t  •  ..  ceieranon 

diagram,    we 
have   along    the   x   axis. 

d'yg        ^djO.a  ,dQ,a 

r~z  --  drrr)   cos0+adsin0=a,,cos   Q-c  (  —  )   sinQ 
at*  at  at 


,N  ., 

d^dT     slnjZJ~adcos^*acsln®+c^!J  —  ^   cos  Q 

then, 

2  dQ 


acsindcosQ=—  -5—  sinQ-dCrr)    sinQcos0+adsinQsin0+c  (-—  ) 
ct  t  d  L  d  t 

sin2Q 


acsinQcosQ= cosQ-df-—-)    sin0cosQ-ajcos0cosQ-cf~) 

dt2        dt  dt 

cos2Q 

ROTATING  TYPE  CARRIAGE:     The  maximum  reactions 
REACTIONS  ON  TRUNNION    on  trunnions  and  main 
AND  FIXED  AXIS  OF        bearing (fixed  axis  of  rocker) 
ROCKER.  are  at  a  maximum  value  at 

the  maximum  powder  pres- 
sure, and  therefore  we  only  need  to  consider  these 
values  in  determining  the  strength  of  parts.   The 
powder  reaction  is  mainly  balanced  by  ths  insrtia 
resistance  offered  by  the  gun  and  the  revolving 
parts.   The  reaction  exerted  on  the  rocker  at  the 
trunnions,  is  that  needed  to  overcome  the  angular 
inertia  of  the  rocker  and  counterweight  which  in 
turn  must  be  equal  to  the  powder  reaction  increases 
the  inertia  resistance  offered  by  the  gun.   There- 
fore the  heavier  the  gun  as  compared  with  the  re— 


930 


volving  parts,  the  smaller  the  effect  of  the  powder 
reaction. 

The  reaction  of  the  main  bearing  is  consider- 
ably augmented  over  that  of  the  trunnion  reactions 
due  to  the  tangential  inertia  forces  of  the  counter 
weight.   The  development  of  the  Cordan  linkage  in 
which  the  rocker  bearing  is  allowed  to  slide  back 
on  a  top  carriage  has  been  largely  to  decrease  the 
reaction  at  the  main  bearing  when  fixed  as  in  the 
revolving  type. 

At  the  maximum  powder  force,  the  recoil 
velocity  of  the  gun  is  small  and  therefore  the 
centrifugal  force  of  the  gun  may  be  negleoted.   The 
tangential  component  of  the  trunnion  reaction,  be- 


T=Pjnaxcos(0-ei)  +  Wsin  9^  —  where 


mjjar* 

and  for  the  normal  component,  N=Praaxsin(0-6  j  )+WgCos 
Therefore,  we  have 

T=proaxcos(ei-ei)(i  --  -i  -  ]+wg 

*r  *cw 
in**  —  +  - 
1  r«  r» 

N=Pmaxsin(0-ei)+Wgcos  Bi     _ 
and  for  the  resultant  trunnion  reaction  S^  s  /  N2+T* 

The  maximum  bending  moment  in  the  rocker  or 
gun  lever  occurs  at  a  section  adjacent  to  the  center 
bearing  of  the  rocker.   This  bending  moment  is  due 
to  the  moment  of  the  reaction  of  the  gun  at  the 
trunnion  minus  the  inertia  moment  of  that  part  of 
the  rocker  above  the  section,  which  is  practically 
one-half  the  mass  of  the  rocker  or  gun  lever. 

The  moment  of  the  inertia  resistance  of  the 

rocker,  becomes, 

-  s 

t    d26    l  .   dta    i  lr   d«s         lr  . 
-  \-  -  =  -  I_  -  =  -  —  r  -   where  —  is  the 
r  dt2         r     «  r«    dt2         r* 

equivalent  mass  of  the  gun  lever  referred  to  the 


931 


trunnions.   The  maximum  bending  moment  at  center 
section  of  the  rocker  or  gun  lever,  becomes, 

t  Xr  d*s, 
M0=vT  -  -  — -  - — -)r    or  in  terms  of  the  maximum 

w     *  r  at 


powder  force 
Ir 


Ar  xcw 
no  +—•+—— 
«  r»  r» 

In  addition  the  section  is  subjected  to  a 
compression,  CO=N+-^  Wrcos  ei=PBaxsin(0-9i  ) +  (Wg*^Wr) 

We  will  now  consider  the  reaction  at  the  fixed  axis 
of  the  rocker  or  gun  lever.   Since  the  tangential 
inertia  effect  of  the  rocker  practically  balances, 
we  will  consider  the  reaction  on  the  main  center 
bearing  as  due  only  to  the  reaction  of  the  gun  at 

the  trunnions  and  tne  inertia  of  the  counterweight. 
The  tangential  inertia  resistance  of  the  counter- 
weight, is 


d«e 

dt* 


Fcwslmq.  - — —   where  q  is  the  distance 


to  any  mass  particle  of 
the  counterweight  measured  from  the  axis  of  rotation 
of  the  gun  lever.   If  rcw  =  the  distance  to  the 
center  of  gravity  of  the  counterweight,  then 

Imq»McwrCB  hence 

d26        ,  .  a  .rcw  Mcw 
FCW=MCW  rcw  Jj#*.wftrtQF*i>-J  — 

It  is  to  be  noted  that  the  point  of  application 
of  Fcw  is  not  at  the  center  of  gravity  of  the 
counterweight,  but  rather  at  the  center  of  percussion 
of  the  counterweight  with  respect  to  the  axis  of 
rotation  of  the  gun  lever.   If  k  is  the  distance 
from  the  axis  of  rotation  to  the  center  of  per- 
cussion, (J2Q 


where  Z  -  the 
/*  +  *         A  +  a 

radius  of  gyration 

of  the  Counterweight  with  respect  to  the  fixed  axis, 

cw  Z2 

therefore  k  =  -   Resolving"  the  resultant 

rc»   reactions  at  the  fixed 


932 


axu  of  the  gun  lever  into  components  normal  and 

alop~  the  axis  of  ths  gun  lever,  we  have,  neglect- 
ing the  centrifugal  forces  as  small, 


Y=N+(*cw+Wr)cos  ®£    or  substituting  values  for 

N,  T,  and  Fcw, 

r 


cw 


X=Pjnaxcos(0-9i)(H-  -  ~-^  -  )+(Wg+Wcw+Wr)sin  6i 


Mcw  —  •  ^ 


Y*pmas 

and  for  the  resultant  we  have,  SQ 

From  these  equations,  it  is  easy  to  see,  that  the 
reaction  at  the  fixed  axis  is  increased  over  that 
at  the  gun  trunnions  by  the  tangential  inertia  of 
the  counterweight  ^  P^cos  (0-6  A 

r  "  "   IM 

With  a  heavy  counterweight,  this  term  is  larger 
and  the  bearing  load  at  the  fixed  axis  becomes 
very  great  with  large  guns.   To  reduce  this  re- 
action and  consequent  weight  of  members,  etc., 
the  Cordan  linkage  disappearing  carriage  developed 
by  Buffington.Crozier  and  the  Krupp  linkage 
have  been  used  for  the  larger  gun  mounts.  Sub- 
tracting, we  have, 

-—^  sinQ-  r-r^cos  Q+d(—)*sin(0-Q)  +adcos  (0-Q)t-c(-—  ) 
dta        dt2         dt  dt 

d*xg     d2yg 

Substituting  the  values  of  -  and  -  ,  we  have 

dt2      dt2 


(a+b)[cos   6  -  sin  6  (^)«]sinQ-b[  sine  ll|+cos(li)t 

dt*  dt  dtz  dt 

cosQ+d(~)asin(0-Q)    +adcos(0-Q)+c  $-)*   =0 
dt  dt 

Expanding  and  simplifying,  we  find 


933 


d*0  d6   2  da0 

a[cos6  — -  -  sin  6(--)    ]  sinQ-b[sin(6-Q) — - 

at  at  dt2 


dt 

,dQ.«        I2      cosM0-B)      ,d0xa 
and    I—*)      =  — •    — — — —     (— • ) 
dt  c*     cos*(0— Q)       dt 


hence 

-:»ni?f  Jsap*   sdj 


ad 


b[sin(e-Q)— +co«<^Q)(-j7)*]-a[cose  S-t  -  sin 
dt  at  dt- 

cos(gf-Q) 


()]    sin  Q  *      -- 


-*-t-h 


'ft*'  » 


...  ^_pk  t  55-  V    c 

;-'l^    cos8  (0-8)1  d6   , 

sin(0-Q)+  —     ,'      ^.   >  (— ) 

c        cos2(0-B)J  dt 


f  -*.  ,-:        : 


Combining   the   acceleration   and  velocity   terms,    we 
have 

[bsin(9-Q)-a  cosSsind]      d26        <{a   sin  Qsin   Q+b 
a ,   *  — —  +     > 

cos(0-Q)  dt*  cos(0-Q) 

f*f^>d«  Jliii  ^(iSJSo     --—  -•      ^    i) 

i*  i 

cos(e-Q)   -  — rtan(0-ft)cos(0-B)-sin(0-B)]    sin(0-Q)> 

AX  * 


d9   «        1^     cos2(0-B)          d9   2 
dt  c        cos2 (0-B)          dt 

Therefore   the   angular   acceleration  of   the   gun,    be- 
comes, 


A     — 
ad          d  dt2' 


dt2        d        d   cos(0  -   Q) 


(rad/sec.2 ) 


834 


where  Ad»  b  sin  (9  -  Q)  -  a  cos  9  sin  Q 

Bd  -   a  sin  9   sin  Q+bcos(6-Q) [tan(0-Q) 

d 

cos(0-B)-sin(0-B))asin(£J-Q)    +  —     cos2(£KB) 

c        cos*(0-Q) 
d   Q 
For   the   acceleration   ac  =   c  -^—  we   eliminate    arf    in 

dt* 
the  equations:- 


dtt.cos£H3(— )au dt< 

-accosQcos0 

d*yg       d0  a       dQ  2 

=  TTT-  sin0-D(-~)  sin*0-c( — )  cosQsinfl 
at         dt         dt 


-acsinQsin(? 


Adding,  we  have 


,d0  2     dQ  a 

in0-d(— )   0-c(— )  sin(0-Q)-accos(0-Q)=0 
dt*      at*        dt       dt 


hence 
*c  3      cos(0-Q) 


g    g     a         a 

Substituting  for  ,  0  ,  ,  .,  (--—  )   and  (——  )  we  obtain 
dtz   dt2  dt        dt 


dt2        dt  dt*      dt 

cos(0-Q) 


cos(0-Q) 


eosg(0-Q)    dt 


935 


Combining,    we    have,  r  i 

«        /y  u         /*.   /*\  4bsin (6-0)+acos0sin9+- 

acos6eos0+bcos(e-0)     d*e  I  d 


cos(0-Q)  dt» 


[tan(0-Q)cos(0-B)-sin(0-B)]*+— 


cos(0-Q) 


c  cosa(0-Q)J 

dt 

d*Q 
Therefore  since  ac  =  c  ,  the  angular  acceleration 

dt*   of  the  gun  lever, 
becomes,  2 

d*Q         5  dt  a 

i   160I.-3V 


d      c  cos(0-Q) 

where  A.,  =  a  cos  9  cos  0  +  b  cos  (6-0) 


r 

=  -  •sbsin(9-0)  +  a  cos0sin9+-—  [  tan(0-B)-sin 


c   cos*  (0-0) 


RECAPITULATION  07  VELOCITIES  AND 

ACCBLBRATIOHS  IH  A  CORDAH  LIHKAGB 
DISAPPEABIN6  GUN  CARRIAGE: 

Let  a+b  =  total  length  of  gun  lever 

a  =  distance  from  cross  bead  to  top  carriage 

trunnion 
b  =  distance  from  top  carriage  trunnion  to 

gun  trunnion 
d  =  distance  from  gun  trunnion  to  elevating 

arm  trunnion  measured  along  gun. 
c  =  length  of  elevating  arm. 
3  =  angle  gun  lever  makes  witn  vertical 
0  =  angle  turned  by  gun 
Q  =  angle  elevating  arm  makes  with  vertical 

M 

—  =  angular  velocity  of  gun  lever 

dt 


936 


£  =  angular  velocity  of  gun 
at 

dQ 

—  »  angular  velocity  of  elevating  arm 

d«9 

•—  •  =  angular  acceleration  of  gun  lever 

d»0 

TTT  z  angular  acceleration  of  gun 

d*Q 

777-  s  angular  acceleration  of  elevating  arm 

dxg 

-—  •  =  horizontal  linear  velocity  of  gun  at 
trunnions 


-  —  =  vertical  linear  velocity  of  gun  at 
trunnions 


a   =  horizontal  acceleration  of  gun  at  trun- 
nions 


,  s   =  vertical  acceleration  of  gun  at  trunnions 

at 

dx 

—  =  velocity  of  top  carriage 
at 

d»x 

-—  -  *  acceleration  of  top  carriage 

dt* 

dy 

—  *  velocity  of  counterweight  and  crosshead 
at 

d»y 

TT7  =  acceleration  of  counterweight  and  cross- 

u  t 

bead 

Then,  in  terms  of  the  angular  velocity  and  acceleration 
of  the  gun  lever, 

(a)  The  velocity  and  acceleration 

of  top  carriage,  are 
dx          d9 

-  -  •  «».  e  -   (ft/iee) 

d*x          da8          dfi  2 

-  =  a  cos  6  --  a  sin  Q(  —  )      Cft/seca) 

dta          dtz          dt 


937 


0>)     The  velocity  and  acceleration 
of  tbe  top  carriage,  are 


"  ~  a  sin  9  —•    (ft/aec) 

at 


=  -  a  sin  9  ~j  -  a  cos  6  (2)'    (ft/sec«) 


(c)     The  velocity  and  acceleration 
of  the  gun,  are 


—^  =  (a+b)cos  9(4r 

dt  at 


-    =  (a+b)cos  0  ^Ii2_ 


dyg 

— 
at 

d*y 
-— 
at 


de 

b  sin  6  —- 
at 


d9  s 

=  b  sin  e  —T+b  cos  6  (-—  ) 
atz         dt 

i[tan(0-Q)cos(0-B)-sin(0-B)]— 
d  dt 


(ft/sec) 

(ft/sec«) 

(ft/sec) 

(ft/sec«) 

(rad/sec) 


df 


(rad/sec«) 


where  A  =  b  sin(9-Q)-a  cos  9  sin 


B  =  a  sin  9  sin  Q  +  b  cos(9-Q)  --  [tan(0-ft) 


cos(0-B)  -  sin(0-B)]in(e>-Q)  + 

c   cos2  ( 

(d)     The  velocity  and  acceleration 
of  tbe  elevating  arm,  are 

dQ   1  cos(g-B)   d9 

=  (rad/sec) 


cdt 


dt2     c  cos(CT-O) 

where  A_  =  a  cos  9  cos  0  +  b  cos  (9-0) 


12 

Bc  =  ~  "J  b  sin(8-0)  +  a  cos  $  sin  9  +  —  [tan(0-Q) 
L  c 

..a    1*   cos2(0-B)l 

cos(0-B)-  sin(0-B)J   + —  «/*  »\  r 

«  c   cos2 (0-Q)  J 

Coordinates  of  the  system: 

Displacement  of  top  carriage  =  x 
Displacement  of  counterweight  =  y 

Distance  from  instantaneous  center  of  gun 
lever  to  gun  trunnion: 


1  =  Aa+b)2cos2  6+b2sin2  6 


COORDINATES  OP  THE  CORDAN  LINKAGE    In  estimating 
DISAPPEARING  CARRIAGE.  the  work  done  by 

the  various  weights 
and  resistances 
during  the  retard- 
ation period  of  the  recoil  it  is  necessary  to  com- 
pute the  various  displacements  of  the  parts  of  the 
system  in  terms  of  the  independent  coordinate 
of  the  system. 

Prom  the  diagram,  to  determine  VI  and  Q  in 
terms  of  the  angle  9  made  by  the  gun  lever  with  the 
vertical,  we  have 

xo  =  (a+b)sin  6+d  cos  0-c  sin  Q 

y0  »  -  b  cos  ©  +  d  sin  0  +  c  cos 
which  may  be  written, 

d  cos  0=  xQ-(a+b)  sin  6+c  sin  Q 

d  sin  0=yo  +  b  cos  6-c  cos  Q 
Squaring  and  adding,  we  have  d*=[xo-(a+b )sin  9]" 

+2[x0-(a+b)sin  9]c  sin  Q  +(yo+b  cos  9  )%2(y0+bcos  9) 

c  cos  Q  +  c* 

This  equation  may  be  put  in  the  form, 


939 


[xQ-(a+b)sin  9]sin   Q+(y0+bcos  9)cos   Q 
[x0-(a+b)sin  9)2+(yo+bcos  9)2 

2 

hence   m   sin    (A+Q)   =  J   "S  da7C2  +  [xo-(a+b)sin  9)%(yo+bcos9  )*> 
where   m  *  /[xQ-(a+b )sin  9]2+(yo+bcos   9)a 

_.    xft-(a+b)sin  9 
A  =   tan      [-  ° 


yo+bcos  t 

From  this  equation  we  may  solve  for  Q  in  terms  of 
9,  and  substituting  in  either  equation  below, 

cos  0  =  3-  [x0-(a+b)sin  9+c  sin  a] 

a 

sin  0=5  lyo+k  cos  ®  ~  c  cos  ^ 

we  may  then  calculate  the  value  of  0  in  terms  of 

the  independent  variable.  Further  if, 
Displacement  of  top  carriage  -  x 
Displacement  of  counterweight  =  y 

The  distance  from  instantaneous  center  of 
gun  lever  to  the  gun  trunnion,  is 

1  =  /(a+b)acos2  9  +ba  sin2  6 

8EACTION3  ON  THE  PARTS  OF     Considering  the  re- 
CORDAN  LINKAGE.  actions  on  the  gun,  it 

will  be  assumed  that 
ths  center  of  gravity 
is  located  at  the  gun 

trunnion.   The  gun  is  subjected  to  a  translatory 
acceleration  divided  into  horizontal  and  vertical 
components  as  well  as  an  angular  acceleration  due 
to  the  reaction  of  the  elevating  arm.   Let 

(1)  PI,  =  the  powder  pressure  along 

the  axis  of  the  bore 

(2)  X  and  Y  =  the  horizontal  and 

vertical  reactions  at  the 

gun  trunnions. 


940 


(3)  W  s  the  weight  of  the  gun  acting 

through  the  gun  trunnion 

(4)  nu*-*  and  md  *  ?  =  the  inertia 

8  dta 

components 

along  the  horizontal  and 
vertical  axis 


(5)  *gT~7  *  the  iner*ia  angular  re- 

sistance 

(6)  X"'and  Y'"=  the  horizontal  and 

vertical  components 
exerted  by  the  ele- 
vating arm  on  the  gun 

Then  for  the  motion  of  the  gun,  we  have 
daxg 

-  -—  -  x"'=  0 
c  d  t 


Pbsin0-Y-m- 

dt« 


yMIdco80-X'"d 

dt* 

For  the  elevating  arm,  we  have  (X1  '  'cosQ+Y1  '  'sinQ)c- 

d«Q 

Ic  —  -  =  0 
5  dt2 

where  Ic  =  the  moment  of  inertia  about  its  fixed 

axis. 

Combining  with  the  moment  equation  of  the  gun,  we 
have 


cos  0"1*  d~F  c  sin  Q 

(Ibs) 


cd   cos(0-Q) 

d2Q 
Ic—  d   sin0*Ig  —  c   cos   0 

Y"'=  -     (Ibs) 
cd   cos    (0-a) 

Next,  to  obtain  the  reactions  X  and  Y  we  must  consider 
the  dynamical  equations  of  the  gun  lever.   By  taking 

moments  about  the  instantaneous  center  of  the  gun 


941 


lever,  we  eliminate  the  unknown  normal  reactions 
of  the  constraints  of  the  carriage  and  counter- 
weight. 

Then  for  moments  about  the  instantaneous 
center  of  the  gun  lever,  we  have 
X(a+b)cos  9+Y  b  sin  9-X'a  cos  9~Y"a  sin  8  -  ra 

d'x  d*9  d'x 

—  a  cos  6  .  IR  —  =  o   where  X'-R*mc— 

R  s  the  hydraulic 

brake  reaction 
on  the  carriage  "mc". 

day 

Y"»mcw  —  —  +wcw  mcw  and  wcw  »  mass  at 

dt* 

weight 

of  the  counterweight.  Combining,  we  have  the 
dynamical  equation  of  the  motion  of  the  disappearing 
gun  carriage  during  the  powder  pressure  period,  as 

follows: 


d«Q       d*0 
—  —  •dcosfl-I-—  —  csinQ 
cdta       Sdta 

-  -    -  ^] 
cd  cos  (0-Q) 


c  cos 


cos  9 


[Pbsin  0-mc   --  J   b   sin  9 

dt8  cd  cos(fr-d) 


dax  day 

-(R+ra  -  )  a  cos  9  -  (mcw  -  +  wcw)  a  sin  9 

dt«  dt« 


d»9 

-  raB  -  a  cos  e  -  ID  -  *  0  * 

dt*  dt2 

For  a  solution  of  this  equation  we  must 
substitute  for  the  various  accelerations  their 
value  in  terms  of  a  function  of  tne  acceleration 

d*6 

-  .  The  hydraulic  brake  resistance  R  may  readily 

dt*   be  obtained  by  considering  the  energy  equation 


942 


of  the  linkage  to  its  recoiled  position. 

If  Ag  3  kinetic  energy  of  gun  at  end  of  powder 

period   (ft/lbs) 
Ac  »  Kinetic  energy  of  top  carriage  at  end 

of  powder  period   (ft/lbs) 
Ae  =  kinetic  energy  of  elevating  gun  at  end 

of  powder  period  (ft/lbs) 
AR  *  kinetic  energy  of  gun  lever  at  end  of 

powder  period   (ft/lbs) 
Aw  ~  kinetic  energy  of  counterweight  at  end  of 

powder  period 

Then  for  the  kinetic  energy  of  the  gun,  we  have, 
if  1  -  the  distance  to  gun  trunnion  from  the 

instantaneous  center  of  gun  movement,  and 
k  radius  of  gyration  about  center  of 
gravity  or  trunnions  of  the  gun. 

2    ) 


d  J  dt 

(ft/lbs) 
For  the  kinetic  energy  of  the  elevating  arm. 

••-%  BOO  •*$  } 

»  _   1*   eos*(0-B)   d6 


c2   cosa(e-Q)   dt 
For  the  kinetic  energy  of  the  gun  lever,  if  kE  » 

*    NOTE:      If  the  path  of  the  sliding  carriage 

has  an  inclination  to  tha  horizontal  equal  to  angle 

d,  then  for  the  equation  of  tho  gun  lever,  we  have 

a2* 

X(a+b)coe  O  +Y(b  sin  O  -  a  cos  0  tan  d  )  -  (  R  +  m  c  -  ) 

dt2 


Substituting  the  values  of  X  and  Y  as  in  the 
previous  equations,  we  have  the  general  dyna»ioal 
equation  of  Motion. 


943 


radius  of  gyration  about  the  center  of  gravity  of 
tbe  gun  lever,  we  have 

A 


R 

If  tbe  top  carriage  and  sides  are  inclined  plane 
making  angle  «  with  tbe  horizontal. 

' 


9(l+tana  V./TB.B. 

dt 

For  tbe  kinetic  energy  of  the  top  carriage 
Ac=rl"ca*cos8  e^TT^   for  horizontal  plane  and 

Ae  =  7  meaacos2  9(l+tan*  o)( — )*  for  inclined  plane 

dt 

For  the  kinetic  energy  of  the  counterweight  and 
cross  bead,     t 

Aw  =  I  mwa2  sin*  9("cTt^a 

When  tbe  sliding  carriage  rides  an  inclined  plane 
the  kinetic  energy  of  the  counterweight,  becomes, 

1  2   d  0  2 

Aw  =  -fflwaa(sin  9+cos  9  tan  <*)  (— *-) 
From  the  principle  of  energy, 


where  WB  ~  work  resisted  by  the  recoil  brake 

WCB  =  work  resisted  by  the  weight  of  the 

counterweight 
Wg  =  work  done  by  the  weight  of  the  gun 

We  =  work  done  by  the  weight  of  the  elevating 

_    arm 

WR  =  work  done  by  the  weight  of  the  gun  lever 

Wc  =  work  done  by  the  weight  of  the  sliding 
carriage 

During  the  powder  period,  the  sliding  carriage  moves 

a  distance  E  and  tbe  gun  lever  angle  increases  from 
60  to  9j  .  The  length  of  recoil  =  L  and  the  re- 

coilad  position  of  the  gun  lever  makes  an  angle  6 
with  the  vertical. 


944 


Work,  resisted  by  the  recoil  brake  »  WB    jf  R=the 
brake  resistance,  then  for  the  work  of  the  recoil 
brake  during  the  retardation  period,  we  have 

WB  »  R(L-E)  (ft/lbs)  where  obviously  L-E»a(sin  62-3in 
and  with  an  inclined  plane  sliding  carriage, 

Work  resisted  by  the  weight  of  the  counterweight  *  W 


cw 


"cw  ~  *w  yw          "w  *  "eight  of  counterweight 
where  yw  =  a(cos  9t-cos  9a)  and  with  an  inclined 
plane  sliding  carriage  yw  =  a(cos  6t-cos  6a)+L  sin  a 
Work  due  to  the  weight  of  ths  gun  =  Wg 

Wg=  "g^g  "here  yg  =  (a+b)(cos  9t-cos  &2)  and  like- 
wise with  an  inclined  plane  sliding  carriage 
yg=(a+b)(cos  6t-cos  92) 

Work  due  to  the  weight  of  the  sliding  carriage  and 

gun  lever  =»  Wr+W,.  ,f  Assuming  the  center  of  gravity 
of  the  gun  lever  at  the  gun  lever  trunnion,  the 
center  of  gravity  of  the  gun  lever  has  the  sane 
displacement  as  the  sliding  carriage.  Hence 

W.+W  *(wr+wc)yc  where  ye  »  (L-E)sin  a  »  — * — (sin6   - 

1   **  COS  ** 

sin  BI).    Hence  when  the  plane  is  horizontal  no 
work  is  done  by  the  weights  of  the  gun  lever  or 
sliding  carriage. 
Work  due  to  the  weight  of  the  elevating  arm  =  We 

where  ye  =  de  (cos  C^-cosQ,) 

de  =  distance  to  center  of 

gravity  from  fixed  axis 
of  elevating  arm. 


=   sin 


in-1     4 


{dg-c»+[x0-(a+b)sin  QI]' 


2  /[x0-(a+b)sin  eja+(yo+b   cos 


946 
(a+b)  sin  9 


-  tan"     [-—  -  -  -  -] 

yg+bcOsQj^ 

' 

[{da-ca+[x0-(a+b)sin  8   l»+(y  o+bcos  6 
Qa  »    sin"1   «j  - 

L     2/[x0-(a+b)sin  9a  1»  +(yQ+bcos   6^  ) 

xn-(a+b  )sin  6 

-  tan"1   [-2—  -  g  -  •] 

yo+bcos  92 


EQUIVALENT  MASS  OF  CORDAN  LINKAGE.     During  the 

powder  period, 
it  is  convenient 

to  express  the  dynamical  equation  of  recoil  in  terms 
of  the  external  moments  or  forces  and  the  equivalent 
mass  of  the  system  tines  the  acceleration  of  the 
coordinate  considered.  The  equivalent  mass  and 
corresponding  reactions  may  be  referred  as  a  function 
of  the  angle  made  by  the  gun  lever  witb  the  vertical 
or  as  a  function  of  the  displacement  of  the  slid- 
ing carriage. 

(1)     Equivalent  mass  referred  to  angle 
"6"  of  gun  lever  witb  vertical  :- 


From  the  dynamical  equation 

of  recoil  for  the  Cordan  linkage  previously  derived, 
we  have,  for  moments  about  the  instantaneous  center 

of  the  gun  lever,  Phla  cos  9  cos(9-0)]-Ra  cos  9H0 

* 


a   sin  9   »  m-    [  (a+b  )cos  9  -  £+   0   8in  9          ft]    +*  a 

dt»  dt« 


cos  9  —  +  mpw   a  sin  9  —  —  +    I 


dta  dt«  dt«  c   cos(0-Q) 

daQ  [bsin(9-Q)-a  sin   Q-cos  9]    d20 

dT2*  +I^  d  cos(Gf-O)  dta 


Neglecting    the   centrifugal  components   of   the 
accelerations,    as   small, 

d*x  a  d«6  d*xg      .  .  da9 

-  -   a  cos  9  -   ;        -  a  =(a+b)   cos  9  - 
dt«  dt8  dt«  dt* 


946 

day  d*9  d*ytf          d*9 

»  a  sin  9  — —  :      •  *  b  sin  9  — — 

dt*  dt*  dt«           dt* 


d*Q  1  daO 

TT  *  ,~  ^    ta  c°s  e   cos   0+fa   °os    (9-^)]-— r 

dt*        c   cos(Qf-Q)  dt* 

Substituting,    we   have  Pj,la  cos   9cos0+bcos (9-0)]- 

Ra  cos   9-ViLIBa  sin  9  = 


-a  sinQcos9]*l 
«cosa(0-Q)      \ 


(a*+2ab)cos*   9+b*] +mca*cos* 

[acos  9   cos   0+b   eos(9-0)]«        ^          [bsin(9-Q>- 
cacos«(Br-Q)  g 


de  _   d«9 

— •  Thus  the  equation  is  in  the  form  of  APh-BR-CW.=D-— 
dt*  dt 

•  here  A  =  a  cos  9  cos  0  +  b  cos  (  9  -  (?) 

B  =  a  cos  9 

C  =    a   sin  9   and   for   the   equivalent   mass    "D" 

D  *  njg[(aa+2ab)cos*   9+ba  ]  +mca*cos*9+mcwa2sin*9+Ir 

[a   cos   9cos0+b   cos-(9-0)]a  [bsin(9-0)-asinQ 

+  I  *  I  

c*cos*(6-0)  d2cos2(0-0) 

cos  9]* 

-^— —     For  the  solution,  during  the  powder 

period,  we  express  the  powder  re- 
action as  a  function  of  the  time,  then 


rlfl  fl^  h  DC\TOn«w  A 

UO  m  U  x  W  «  .  A  XX   »f      > 

_./_«.  (_ )t    ,  _  MS  -(_ ,t 

where  Vf  is  the  velocity  of  free  recoil  of  the  gun 

Integrating  again,     A        BR+CWCW 

a  _  _  u   TP  _  f         \  +2 

•  D  ^  *    (  -  20   ) 

where  E  is  the  displaceaent  of  the  gun  in  free  re- 
coil. From  the  solution  of  these  equations,  we 
obtain, 


947 


et*eo*   Mge  ~  <~  -  >T*  "here  T  =  powder  interval 


BR+CW 


The  angular  displacement  and  the  angular  velocity 
of  the  gun  at  the  end  of  the  powder  period.  Sub- 
stituting these  values  in  the  energy  equation,  we 
have 

*B  *  Ag*M*R  +Ac+Aw+Wg+Jre+iR  +*C-WCW  and  there- 

fore can  readily  determine  R  the  total  braking 
resistance. 

(2)     Equivalent  mass  referred  to  dis- 

placement of  sliding  carriage  X;- 

In  place  of  a  movement  and 

angular  acceleration  equation,  we  nay  consider 
the  inertia  and  the  reactions  of  the  system  as  re- 
duced to  an  equivalent  translatory  mass  and  force 
as  a  function  of  the  displacement  of  the  sliding 
carriage.  Therefore  reducing  the  motion  of  the 
system  to  one  of  translation  along  the  path  of 
the  top  carriage. 

By  direct  analysis,  we  have,  if 

Pjj  *  powder  reaction 

X  and  Y  =  components  of  gun  trunnion  reaction 

XI  and  Y'  =  components  of  top  carriage  on  gun 

lever 
X"  and  Y"  =  components  of  reaction  on  gun 

lever  at  crosshead 
X'11  and  Y1  '  '  =  components  of  elevating  arm 

reaction  on  gun 
mg  =  mass  of  gun 
mcw  =  mass  of  counterweight 
T.S  =  mass  of  elevating  arm 

ro_  -  mass  of  top  carriage 
c  r      • 

mB  =  mass  of  gun  lever 

K  e> 

Then,  for  moments  about  the  instantaneous  center 


948 


d*x 
X(a+b)cos  6+y  b   sin  6-Y"  a  sin  9-mR  —  a  cos  6-Ir 

dt« 


-X'  a  cos  6  =  o 
dt» 

Dividing  through  by  a  cos  9,  we  have 


)y  tan  e-Y-tan  e  -.„ 

a    2  R  dt*    a  cos9 


Since 

d20 


sin  0 


cd  cos  (0-G) 

d20 

cos 


cd  cos  (0-Q) 


ire   have   on   substitution, 

P^(  -  cos(?+-tan   6   sin0)-Wcwtan  6-X1      as    the  equi- 

3  8 

valent  force  acting  along  the  top  carriage  guides 

and  _   d2Q  J  _   d»C 

Ir—  rr  d    cos0-Iflr—  re    sin 

b 

t 


dta      a         *dt2      a  cd  cos 


a        sin        JtfTa   c  cos       v 
a+b  °dta  gdt2  b  .          d*y 

-         -  9+       - 


tan 
a       cd  cos  (fif-Q)  ""dt* 

Ir     4*9 

+  — — — —  — —  as  the  equivalent  inertia  resistance 

a  cos9  dt»  offered  by  the  mass  of  the  total 

system  reduced  to  the  path  along  the  top  carriage 
guides. 


949 


THROTTLING  CALCULATIONS  WITH  ADD  WITHOOT 
A  FILLING  IM  BPFF1B. 

4.7  Gun  Trailer  Mount  with  U.  S.  Variable  Recoil 
Valve. 

w  =  weight  of  projectile        45  Ibs. 

v  =  muzzle  velocity  2400  ft/sec, 

v  =  166.13  (in.) 

w  -  weight  of  ponder  charge      11  Ibs. 
pb  max  *  34000  Ibs/sq.in. 
b  =  36"  (0°  to  45°) 

4.7  (0°  to  45°) 

X  =   total   resistance   =  17806.9706  Ibs. 

*r  *    weight  of  recoiling  parts  =   7560  Ibs. 
St  3  spring   load  * 

Pv£=Sf  3    spring   load  at  end  of  recoil  »  16140   Ibs 
Fvj  =  So   =  spring   load  at  assembled  height  = 
Wr  *    1.3  =   9800    Ibs. 

16140-9800        6340 
St  *    So  "  ~36 "   "ST  "   176-U1   lb8   " 

increase  of  spring  load  per  inch  of 
recoil. 

«  *   maximum  angle   of  elevation  *   45 ° 
Wr   sin  a  =  weight   component  =    7550  *    .70711.= 

5338. Ibs.-  6805 

Bg  =  stuffing  box  friction  *  2,25  *  100  *  225 
Rg  =  guide  friction  =  Hru  cos  » 
u  =  coefficient  of  friction  «  .15 
Rg  =  7550x.70711x.15  -800.8021 

Effective  area  of  recoil  piston  =  9.337  (sq.in) 


950 


METHOD  OF  PLOTTING  VELOCITY  CURVE  -  VARIABLE 


RESISTANCE  TO  RECOIL. 


Kt, 


'P  «  Vfo  -  —  ;  (ft/sec) 


2M 


(ft) 


When  projectile  leaves 
the  muzzle. 


Kt. 


(ft/sec) 


2M 


(ft) 


The  maximum  restrained 
recoil  velocity  and 
corresponding  recoil. 


ob 


where  Vfm=Vfo  *  —  (tm-to)[l-  _^l__^_r_ 
Mr  4Mr(Vf-V0) 


'ob 


6Mr(Vf-V0) 


](t.-t0)  (ft) 


and         K(T-t0) 

t.  -  T  -  -= —  (Sec) 


Kt 
VP-Vf  --  (ft/sec) 

M 
M 


Kt* 

Er>B  --  (ft) 
2Mr 


>  At  the  end  of  the  powder 
period 


During  the  retardation  period, 
/  2[K-  \  (b+x-2Er)](b-x) 

/  ^ 


and  therefore 


cA 


2[K-  -  (b-x)-2Er)](b-x) 

• 


•(sq.in) 


13.2 


-  Rt  -  W, 


961 


which  gives  the  required  throttling  area  with  a 
variable  resistance  to  recoil  during  the  retard- 
ation period. 

Sf-S0x 

Ph=K+Wpsin0-Rt-(S0+ — )  for  spring  return  re- 
cuperators. 

Equivalent  throttling  area  :  4.7  A. A. Trailer, Model 

1918 
1_    1     1 

W2       \tf  2         U72 

Area  of  one  hole  =  .0113  sq.  in. 
In  battery  -  Wxb=  20  holes  =  .226  sq.in.  w|  =.0510 

Wx  =103  holes  =1.1639sq.in.  w_  =1.3546 
t  *  t 

4"  Recoil  -  W*   =81   holes  =.9153   sq.in.   W*      =      .8377 
xf  xa 

W«    =88   holes=.9944   sq.in.      W^=        .9888 

8"   Recoil  -Wx   =86  holes   =.8781   sq.in.      WS      *      .9566 

\  \ 

WXj  =    77   holes   =    .8701   sq.in.W|t=        .7570 

12"  Recoil  -W,   =98  holes=1.1074   sq.in.   W.     =   1.2263 
xa  *a 

Wx   =67   holes   *    .7571   sq.in.    W|   =      .5732 

i  xt 

16"   Recoil  -Wx   =107  holes  =   1.2091sq.in.W»   =   1.4619 

Wx   =59   holes   =    .6667   sq.in.    W»    =    .4444 
*i  *i 

20"  Recoil-W_  =115   boles  =  1.2995   sq.in.W*   « 1.6887 

*  2  • 

Wx   =50  holes  =   .5650  sq.in.      WJ  =      .3192 
24"  Recoil   -W,   =125   holes=    1.4125   sq.in.W*   =   1.9951 

X8  ** 

Wx   =  30  boles   =   .3390  sq.in.    W«      =   .1149 

28"  Recoil  -Wx  =140   boles  -  1.5820  sq.in.W*    -2.5027 
WX2=30  holes  =    .3390  sq.in.    WJ4=      .1149 

32M  Recoil-Wx   =140  holes   =  1.5820  sq.in.    W$   =2.5027 
t 

Wx   =   24   holes  =    .2712    sq.in.    W|  =      .0735 
36"  Recoil   -Wy  =152   boles=.1.7176  sq.in.Wx     =   2.9501 

*2  2 

Wx   =0  holes  =    0  sq.in.          W*      =          0 


962 


Equivalent  throttling  areas-4.7  A.A.  Trailer, Mpdel 

1918 

-1  -  y,   »•  .  i  ,   y  -*«-« 
i»e  y 

In  battery, i-  •• —  +  — -=20.355,  .049127,  .221 

W»   .1.3546   .0510 

4"  Recoil,^-  =  — +— - — =2.204,      .453720   .675 

we    .9888  .8377 

8-B.oo.il.ij.  7^570  +i956e-  2.366   .422654   .650 
12"Recoil,jL  -  -^  *  j-^f™  2.569,  .389256.  .623 

6 

16»8ecoil  |j  -  -ijjj   *  -j-^ijj  =     3.934      .340831      .683 
20"  R..011..JL.  -ijg  *  T-i^  -      3.724      .2685S8,    .818 

11  1 

24"  Recoil,—  =  + *    9.204      .108648      ,329 

we        .1149       1.9951 

28"Recoil,-—     *  — »4  =        9.102,    .100865,    .331 

6          .1149  2.5027 

32  "Recoil,  —  =  — —  +  — =14.00,      .071428,    .267 

we          .0735       2.5027 

36"  Recoil,  —     

Equivalent    throttling   area  -  4.7   A.  A. Trailer, Mgdel 

1918 
(Calculated) 


R      =  =  W*    =  

175WJ  175Rh 

K  »  —  =   1.43,    Ka=2.045,    A  =  9.337   sq.in.,  A3=913. 994 
.7 

K«A8    =   1869-U7 


863 


i  t  1"  Recoil  •••  *  1869-117<16-°60'  -  .800656,  W...447 

175x12072.42 

A  t  4"  Recoil*  V£  =  1869.117*19.330 

176x11330.17 


____^* 

A  t  12"  Recoil=W|  =  1869'11?Xl6'614  =.315279,WX  =  .561 

175x9350.81 

^B^^^^_^B^2 

A  t  16"  Recoil»W|  «  1869.117x15.090  ,t290878^w  ,  >531 

175x8361.14 

A  t  20"  Recoil=W«  =  1869>11?Xl3>245  =.  254184,  Wx  *  .504 

175x7371.46   . 

1869.117x11.569 

A  t  24"  Recoil=W*  =  -  =.  223998,  Wx  =  .473 

175x6381.79 

•^^vwft 

A  t  28"  Recoil  -W«»  1869.117x9.395  =  <174836^w  =  >418 

175x5392.11 

9 

A  t  32"  Recoil=W»  =  1869'  117x6'604  3  .  105806,  «v  »  .325 

175x4402.43 

A  t  36"  Recoil  - 

Equivalent  throttling  area  -  With  filling  in  buffer 


Aj,  =  1.767  sq.in.    AQ  *  .76  sq.in. 

A  «  9.337  sq.in.,   A-  =  .69  sq.in. 

^-  =  -==»+-=»-^ — +-^—=1.73+2.10=3.83 
wb   .76  .69   .577  .476 


K«(A-Ab)3V« 

Wg  »  -^ 


„   a        -,               1.43*  (9.337-1.767)3xl9.33 
At   4"   Recoil, W|   =   , 


176(11330  -  L33   -1.767  .19.88 


Wa 

"e 


175x.261 
2.04x433.76x373.64 


1.43    (9.337xl.767)3xl8.03 
At   8"Recoil,H|   =  • 


2.04x433.76x325.08 
175(10340-      '      *    1! 


-160'  "«  •  -*00 


A«  12" 


2.04x433.76x275.89 

W  =         "" *" "• ^^ *^^^m—^m^m^* 


1.76.5.51x275.89 


-160'  "«  *  _ 

it  16"  Recoil,  *.  >  1.43'(9.337-1.767)-16.09' 


176(8361  .  1.33  .1. 


175x.261 
a   2.04x433.76x227.7 

175(8361 . 1-76>5-"-^-7 


45.67 


955 


r 

3_nV 


FI6.  A 


FIG.  B 


FIG.C 


FIG.  D 


GUN  LUGS 
PLATE  I 


966 


.  201484_  ,    _138> 
1454775 


COH8TBUCTIVI  DETAILS. 

GUN  LUGS  -         Typical  gun  lugs  are  shown  in 
PLATE  I.        Plate  I,  figures  A,  B,  C  and  0 

respectively.   A  gun  lug  properly 
speaking  is  an  integral  part  of  a 
gun,  being  an  integral  part  of 
the  bree   ring. 

Fig.  A,  shows  an  arrangement  used  on  the  75 
m/m  Futeaux  Model  1897.   Surrounding  the  lug  is  the 
piston  rod  yoke  connecting  the  piston  and  gun  lug, 
fig.  H  -  Plate  3.   The  piston  rod  yoke  and  gun  lug 
are  connected  by  the  key  passing  through  both  lug 
and  yoke. 

Pig.  B,  shows  a  simple  construction  used  on 
the  4.7"  anti-aircraft  mount, model  1917. 

Fig.  C,  shows  the  lug  of  the  155  m/ra  G.P.F. 
gun.   The  two  "holes  in  the  lug  are  for  the 
hydraulic  and  recuperator  piston  rods. 

Fig.  D,  is  a  lug  for  connecting  the  recoil 
sleigh  n  slide  to  the  gun  in  the  155  m/m  Schneider 
Howitzer.  The  sleigh  and  gun  are  further  connected 
by  a  front  clip,  but  the  pull  during  the  recoil 
however,  is  exerted  through  the  lug,  the  front 
merely  supporting  the  gun. 

Two  sections  are  important  in  the  design  of 

a  lug:-  the  section  "ah"  just  above  the  rods,  which 
carries  mainly  shear,  and  the  section  "mn"  at  the 
breech  circumference  which  should  be  designed  main- 
ly for  bending. 

ARRANGEMENT  OF  GUIDES          The  recoiling  parts 
AND  CLIPS.  are  constrained  to  recoil 

ID  the  direction  of  the 

axis  of  the  bore  by  the  engagement  of  clips  attached 
to  the  gun  or  recoiling  mass,  in  suitable  guides  on 


967 


the  cradle  or  recuperator  forging.  The  reaction 
between  the  guides  and  clips  balance  the  weight 
component  of  the  recoiling  parts  normal  to  the 
bore  and  the  turning  moment,  due  to  the  pull  of  the 
various  rods  about  the  center  of  gravity  of  the 
recoiling  parts.  Due  to  the  large  turning  moment 
caused  by  the  pulls  as  compared  with  the  weight 
component  of  the  recoiling  parts  normal  to  the 
bore  and  more  or  less  "play"  between  the  guides  and 
clips,  the  normal  reactions  exerted  by  the  guides 
on  the  clips  are  more  or  less  concentrated  at  the 
end  contacts.  The  distribution  of  the  bearing 
pressure,  of  course,  depends  upon  the  elasticity 
and  play  between  the  clip  and  guides,  and  there- 
fore, assumptions  based  on  experience  must  be  made 
as  to  the  proper  surfaces  required.   In  older 
type  mounts,  we  have  a  continuous  clip  on  the 
gun,  engaging  in  the  guides  of  the  cradle.   Unless 
the  gun  clips  are  sufficiently  long,  we  have  a 
varying, (gradually  decreasing  distance),  between 
the  clip  reactions  assumed  concentrated  at  the  ends 
and  thus  the  friction  of  the  guides  increases  in  the 
recoil.  Due  to  heating  of  the  guides  firing  unless 
sufficient  play  is  allowed  for,  warping  of  the  guides 
may  cause  a  binding  action  between  the  clips  and 
guides. 

Therefore,  due  to  these  considerations, 

(1)  the  increase  in  clip  reaction 
towards  the  end  of  recoil, 

and 

(2)  the  difficulty  of  preventing  warp- 
ing of  the  guides  or  clips 

and 

(3)  the  necessity  of  a  long  gun 
jacket,  continuous  gun  clips  have  been 
nore  or  less  discontinued  in  modern 
artillery. 

When  gun  clips  are  used  we  have  combinations 
of  three  or  more  gun  clips.  When  only  three 


966 


clips  are  used  it  is  possible  to  maintain  practically 
only  two  clips  in  contact  with  the  guides  through- 
out the  greater  part  of  recoil.  This  is  an  advantage 
since  any  warping  of  the  guides,  etc.,  does  not 
materially  effect  the  operation  of  the  recoil.  With 
four  or  more  gun  clips,  we  have  one  or  more  inter- 
mediate clips,  thus  necessitating  a  more  careful 
lining  up  of  the  gun  clips  and  design  of  the  guides 
to  prevent  warping  unless  considerable  play  is  to 
be  allowed. 

Referring  to  fig. (3)  Plate     we  have  an 
arrangement  of  three  clips  A, 8  and  C,  which  recoil 
to  an  intermediate  position  A^B^C1,  where  the 
rear  clip  leaves  the  guide  and  the  front  clip  enters 
the  guide.   If  the  clips  are  equally  spaced  as  they 
should  be,  this  intermediate  position  is  one-half 
the  length  of  recoil.   In  the  final  position  the 
clips  are  in  the  position  A"  B"  C"  at  the  end  of 
recoil.   If  "1"  is  the  distance  between  clips, 
since  A  should  not  leave  the  guide  until  C  enters 
the  guide  and  at  the  end  of  recoil  6  must  be  still 
in  contact  with  the  guide,  the  length  of  guide 
should  be: 

Win.  length  of  guides  =  2L  =  b,  (3  gun 

clips)  and  therefore, 

b 

Distance  between  clips  1  =  - 

2 

With  three  clips,  during  the  first  half  of  the 
recoil,  the  coordinates  with  respect  to  the  center 
of  gravity  of  the  recoiling  parts  of  the  front 
and  rear  clips  respectively,  become  those  of  B  and 
A  while  during  the  latter  half,  they  become  those 
of  C  and  B. 

With  four  clips  we  have  an  intermediate  clips 
always  in  contact  with  the  guides;  hence  a  careful 

alignment  is  necessary  with  more  or  less  to  prevent 
any  binding  action  of  the  middle  clip  and  throw 
the  greater  part  of  the  clip  load  on  the  extreme 


969 


front  and  rear  clips  respectively  .   Referring  to 
fig.  (3)   Plate  (      the  clips  A,B,C  and  D 

move  from  the  battery  position,  to  the  midway 
intermediary  position,  that  is  when  clip  A  leaves 
the  guide  and  clip  D  just  enters  the  guide.   If  "1" 
is  the  distance  between  the  extreme  clips  in  bat- 
tery, i.e.  between  A  and  C,  or  between  B  and  0 
when  clips  are  equally  spaced  as  they  should  be, 
we  have  1  =  b,  that  is  the  distance  between  clips 
equals  the  length  of  recoil.  Further  the  minimum 
length  of  guide  =  -  I  -  ~  b. 

With  four  clips,  the  coordinates  of  the  front 
and  rear  clip  reactions  with  respect  to  the 
center  of  gravity  of  the  recoiling  parts  during 
the  first  half  of  recoil  become  those  C  and  A 
respectively,  while  during  the  latter  half  they 
become  those  of  8  and  0  respectively. 

Let  us  now  consider  the  front  and  rear  clip 
reactions  between  the  guides  and  clips  of  the 
gun. 

The  clip  reactions,  become,  for  the  front 

Clip'     P 


for  the  rear  clip,     Pbs  +  Bdb*«rcos0(xt-nyx  ) 


where  Pb  =  the  max.  total  powder  reaction  on  the 

breech  (Ibs) 
e  =  the  distance  from  the  axis  of  the  bore 

to  the  center  of  gravity  of  the  recoil- 

ing parts  (inches) 
B  -  the  total  braking  pull  excluding  the 

guide  friction  (Ibs) 
dfc  =  the  distance  down  from  the  center  of 

gravity  of  the  recoiling  parts  to  the 

line  of  action  of  the  center  of  pulls 

(inches) 


960 


Wr  =  weight  of  recoiling  parts  (Ibs) 

xt  and  yt  =  coordinates  of  front  clip  reaction 

measured  from  the  center  of 

gravity  of  the  recoiling  parts. 
xa  and  ya  =  coordinates  of  rear  clip  reaction 

measured  from  the  center  of  gravity 

of  the  recoiling  parts. 
n  =  coefficient  of  guide  friction  =  0.15  approx 

0  =  the  angle  of  elevation. 

1  =  xt  +  x  -  the  distance  between  clip  re- 

actions . 
Since  ny  and  ny  are  small  as  compared  with 

x  and  xa  respectively,  we  have  for  a  close  approx- 

imation, 

Pbe+Bdb-Wrcos0.x2  Bdb-Wrcos0   x2 

=     -  •  -    (approx) 


,  n 

1-2  ndr  1 

Pbe+Bdb+Wrcos0  xt     Rdb+Krcos 


t 

(approx) 


l-2ndr 

where  dr  a  mean  distance  from  center  of  gravity 
of  recoiling  parts  to  guide.  The 
guide  friction,  becomes, 

2nBdb+nWrcos0(x  -x. ) 

RI  3  n(Q  +0.)  -  E — —  (Ibs) 

l-2ndr 

The  following  table  is  useful  in  the  layout 
arrangement  for  the  gun  clips  and  proper  length 

of  guides,  as  well  as  showing  the  change  in 
clip  reaction  and  guide  friction  for  ths  two 
combinations . 


961 


M 

M 

X 

X 

1 

1 

X 

X 

^w 

M 

o 

O 

o 

0 

Ll 

Ll           Ll 

t-, 

tit 

•rt              S 

•a 

c 

c             c 

c 

CV3 

IN)                 + 

CO 

(  s  q  T  ) 

Q 

1               -o 

1 

\        Ml/ 

'O^ 

03                  TJ 

jQ 

UOT^OTJjJ       3  p  T  n  £}       T  *  ^  O  J. 

CD 

CD 

C 

C 

•* 

O3 

X 

is 

cs 

M 

X 

O 

S 

O 

w 

Li 

o 

f  s  q  T  ) 

tf 

o 

t, 

uot^aeay     djig     jesg 

ja 

XXI               * 

.£> 

CD 

JQ 

N—  ' 

TJ 

CVJ 

CD 

X 

9 

M 

CO 

X 

0 

^Sl 

o 

8 

L! 

0 

5£ 

o 

(  s  qt  ) 

1 

O                     Li 

£ 

uoj  >o*  aa     <*T  TO     »uoj,g 

CD 

I 

ft 

03 

CO 

psatnbsj 

.O 

sapinS     jo     q^JJuai     *Ufh 

-O                        CO  1    CO 

s  u  o  j  3.  3  p  3  j     d  |  i  o     jeoj     pus 

^  u  o  j  j     u  a  3  u  ^  o  q     aouc^efQ 

^)    1    03                       J3 

q     i  T  o  o  3  j     jo    sujta^     u  T 

sdi-[o     ussii^aq     souF^sta 

XI    1    CM                  X(  1    CVJ 

sdTTO     jo      »on 

CO                          sf 

962 


DESIGN  AND  STRENGTH  OF     In  the  design  of  gun 
GUN  CLIPS  AND  GUIDES.   clips  and  guides,  the 

following  points  should 
be  considered: (1)  General 
considerations  as  to  lay 

out,  protection  from  dust,  etc;  (2)  the  arrange- 
ment of  clips  and  guides  as  outlined  in  the  previous 
paragraph;  (3)  the  computation  of  the  maximum  clip 
reactions;  (4)  the  design  of  the  clip  or  guide 
for  allowable  bearing  pressure;  (5)  the  strength 
of  the  clip  or  guides  at  their  various  critical 
sections,  to  resist  bending,  direct  stress  and 
shear. 

(1)     The  location  of  guides  in  the 
direction  normal  to  the  axis  of  the 
bore  should  be  based  on  the  follow* 
ing  considerations :- 

(a)  From  a  cross  section  of  the 
gun  and  recuperator  forging, 
the  best  position  of  guides 
and  gun  clips  can  be  located 
with  consideration  for 
minimum  stress  in  gun  clips. 
This  requires  that  the  guides 
be  located  as  near  the  axis 

of  the  bore  as  possible. 

(b)  For  constructive  reasons, 
it  is  good  design  to  keep 
the  various  parts  connected 
with  the  recoiling  parts  as 
near  the  axis  of  bore  as 
possible. 

(c)  The  reactions  of  the  guides, 
however,  are  quite  independent 
of  the  position  of  the  guides 
in  a  normal  direction  to  the 
bore,  but  since  the  resisting 
section  of  the  cradle  or  re- 
cuperator forging  is  very  large 


963 


Fk5.  F 


GUN  CLIPS 
PLATE  2 


964 


Q< 


t>j 


0 


-]gu 


u 


•^Bo 


965 


as  compared  with  those  of  the 
gun  clips;  gun  clips  with 
long  projections  downward  from 
the  gun  clip  jacket  due  to 
guides  too  far  below  the  axis 
of  the  bore  are  undesirable. 
Hence  the  location  of  the  guides  depends 
upon  construction  and  fabrication  features  with 
due  consideration  to  the  strength  of  the  gun  clips. 
These  features  in  general  demand  that  the  guides 
be  located  as  close  to  the  axis  of  the  bore  as 
possible. 

(2)  For  small  guns,  three  clips  equally 
spaced  as  described  in  the  previous 
paragraph  should  be  used.   The  front 
and  rear  clips  should  be  bevelled 
off,  so  that  smooth  entrance  may 

be  made  into  the  guides.   Bronze 
liners  either  in  the  clips  or  guides 
should  be  used.   For  larger  caliber 
guns,  more  clips  should  be  used 
since  the  clip  reactions  and  cor- 
responding friction  are  reduced. 

Considerable  tolerance  should  be 
allowed  but  very  careful  alignment 
made  in  order  to  prevent  possible 
binding . 

(3)  The  computation  of  clip  reactions 
has  been  tabulated  in  a  previous 
paragraph,  for  the  common  arrange- 
ment of  either  three  or  four  equally 
spaced  gun  clips. 

(4)  The  bearing  contact  during  the 
recoil  between  guides  and  clips, 
depends  upon  tolerance  between  the 
guides  and  clips  as  well  as  the 
elasticity  of  the  material,  and  on 
the  magnitude  of  the  wear  between 


966 


the  clips  and  guides.  Therefore, 

we  see  the  distribution  of  bearing 
pressure  and  the  length  of  contact 
is  completely  indeterminate.  From 
practice,  however,  the  following 
assumption  will  be  made: 

(a)  Length  of  gun  clip  1  =  1.8d 
(in.)  approx.  where  d  =  diam. 

of  bore. 

(b)  Constant  length  l'=1.5  d(in.) 

(c)  Distribution  of  pressure 
assumed  triangular. 

Therefore,  if  b1  =  contact  width  of  clip  and  guide, 
(inches)  we  have  for  the  maximum  bearing  pressure 
due  to  the  clip  reaction  Q  (Ibs) . 

Q 

1    Ibs.  per  sq.in. 
b  d 

Now  the  max.  allowable  bearing  pressure  steel 
on  bronze,  becomes,  pgm  =  600  to  800  Ibs.  per  sq*in. 

Hence  b'  =  .0017  to  .0022  -  (inches) 

d 

The  distance  1-1 '  should  be  the  bevelled  length 
of  clip  distributed  on  either  end. 

With  eccentric  pulls  the  side  thrust  between 
clips  and  guides  causes  a  bearing  reaction  Q1  and 
if  b"  is  the  depth  of  guides  in  contact  with 

clip,  we  have,  b'  =  .0017  to  .0022-=-  (inches) 

d 

(5)     The  strength  of  gun  clips  depends 
upon  the  form  or  type  of  gun  clip 
used.   In  fig. E, plate  2,  we  have 
the  minimum  bearing  contact  (w-x). 
The  required  thickness  of  the  toe 
T  is  based  on  bending  at   section 
(a-b).  Since  the  front  clip  re- 
action causes  this  bending,  and 
the  load  is  divided  between  two 


L. 


FIG.  H 


967 


FIQ.  K 


, 
±j 


FIG.  L. 


R5TOM   ROD 


GUH 


Pt-ATE  -4- 


966 


front  clips  on  either  side.  We  have, 

,    (»-x)         /Q  (w-x) 
1.225  /— 1 =  0.912  /  — (in) 


where  fm  =  —  elastic  limit  of  material  used. 

STRENGTH  OF  RECOIL  PISTON     The  greater  part 
RODS.  of  recoil  piston  rods 

are  subjected  to  ten- 
sion during  the  re- 
coil, and  com- 
pression during  the  counter  recoil  due  to  the 
counter  recoil  buffer  reaction.   In  a  few  types 
of  recoil  systems,  we  have  compression  in  the  rod 
during  the  recoil,  an  example  being  in  the 
pneumatic  cylinder  of  the  16"  U.  S.  Railway  mount. 
The  critical  diameter  of  2  recoil  piston  rod 
is  at  the  smallest  section  within  the  gun  lug  as 
shown  in  figures  H,  K  and  L, Plate  4.  This  diameter 
should  be  based  on  the  recoil  pull  at  maximum 
elevation  and  the  inertia  load  at  maximum  acceleration 
This  load  is  the  same  that  occurs  for  the  gun  lug. 
Let  P  a  the  total  fluid  reaction  +  packing  friction 

on  piston  and  rod  (Ibs) 
B  =  total  braking  (Ibs) 

Pb  =  total  max.  powder  reaction  on  breech  (Ibs) 
fm  =  allowable  fibre  stress  of  material  used 

(Ibs/sq.in) 

wp  =  weight  of  rod  and  piston  (Ibs) 
wr  =  weight  of  recoiling  parts  (Ibs) 
d  =  diameter  of  smallest  free  section  at  gun 
lug, 


0.7854  f 


m 


For  hollow  piston  rods,  with  a  "filling  in"  or 
spear  buffer  chamber,  we  must  consider  a  section 
the  greatest  distance  from  the  piston  but  passing 
through  the  buffer  for  maximum  inertia  and  minimum 


969 


thickness  of  the  rod.   Let  w'  =  weight  of  piston* 

rod  to  section  (Ibs) 

dro  =  outside  diam.  of  buffer  rod  (in) 
drj  =  inside  diam.  of  buffer  chamber  (in) 
Then  using  the  previous  symbols,  we  have, 


dro  ~dri  =  -  usually  dri  is  fixed 
0.785  fm       in  consideration  of 

the  buffer  design,  hence 
dro  is  determined  from  the  above  formula. 

When  piston  rods  are  subjected  to  compression, 
during  the  counter  recoil  or  with  a  pneumatic 
recuperator  during  the  recoil,  the  rod  should  be 
treated  as  a  column  loaded  and  constrained  at 
both  ends. 

The  maximum  column  load  on  the  rod  equals  the 
maximum  counter  recoil  buffer  load,  which  may  be 
roughly  estimated  on  the  basis  of  counter  recoil 
stability  at  horizontal  elevation.  If 

Cg  =  constant  of  counter  recoil  stability  = 

0.85  to  0.9 

Ws  =  weight  of  total  gun  +  carriage  (Ibs) 
lg  =  distance  from  wheel  contact  to  line 
of  action  of  Ws,  recoiling  parts  in 
battery  (in) 
b  =  height  of  center  of  gravity  of  recoil- 

ing parts  above  ground  (in) 
Bj{  -  counter  recoil  buffer  reaction  (Ibs) 
Fvi  =  recuperator  reaction  in  battery  (Ibs) 

R'=  approximate  total  friction  (Ibs)  =  0.3  W_ 
W  1  '  W  1  ' 

then  BjJ  +R'-Fvi=Cs  -2-i   from  which  BJ=Fvi+C  '-£-£-  -  R1 

h  h 

(Ibs) 

thus  giving  the  maximum  compression  load  on  the 
rod. 

With  pneumatic  recuperators  if  the  rod  is 
under  compression,  the  maximum  compression  is 


970 


x,- 


«•*- X, 


FIG.   M 


kM-4H 


JLL 


FKa.    M 


RECUPERATOR      FORGlhSS 
Pt-ATE     5 


971 


liable  to  be  either  at  the  beginning  or  end  of 
recoil.  At  the  beginning  we  have  the  initial 
recuperator  reaction  +  the  inertia  load  of  the 
rod,  and  at  the  end  of  recoil  the  maximum  recuperator 
reaction. 

TRUNNIONS  AND  SUPPORTING     In  older  mounts,  the 
BRACKETS.  trunnions  were  an  integral 

part  of  the  gun,  the  gun 
setting  directly  in  the 
top  carriage.  With 

mounts  using  a  recoil  system  between  the  gun  and 
top  carriage,  the  trunnions  are  usually  bolted 
by  a  supporting  bracket  to  the  cradle,  though  when 
the  recuperator  becomes  a  guide  support  replacing 
the  necessity  of  a  cradle,  the  trunnions  often  are 
an  integral  part  of  the  recuperator  forging. 

Plate  4  shows  recuperator  forgings  with 
trunnions  an  integral  part  of  the  forging,  figures 
M  and  P,  while  fig.  N  shows  a  recuperator  forging 
with  a  trunnion  bracketed  on. 

Plate  6  shows  typical  trunnions  and  their 
supporting  brackets  which  are  bolted  to  cradle  . 

In  fig.  M,  consideration  only  of  the  design 
of  the  trunnion  itself  is  necessary,  while  in 
fig.  P  the  strength  of  section  m  y  should  be 
considered  as  well.  Section  mn  is  subjected  to 
bending  and  shear  combined  with  direct  stress. 

DESIGN  Or  TRUNN10KS: 


Let  w  =  bearing  length  of  trunnion 
Of  3  outside  diam.  of  trunnion 
d*  *  inside  diam.  of  trunnion  at  section  "mn" 
f  =  max.  fibre  stress,  -  Ibs.  per  sq.in. 
ft,  =  allowable  bearing  pressure  -  Ibs. per  sq. 
in. 


972 


Let  w  =  width  of  section  "ab"  just  above  the  rods 
w1  =  width  of  section  "mn"  at  the  contact  of 

breech  circumference  and  lug. 
dfc,  =»  the  distance  down  from  "mn"  to  center  of 

gravity  of  pulls 
d  -  depth  of  lugs 
T  =  longitudinal  length  of  lug 
Pb  -  max.  total  powder  pressure  on  breech 
nc  =  weight  of  recoiling  parts  attached  to 

lug. 
wr  =  total  weight  of  recoiling  parts 

Then        w 

B*  r<pb-B) 

wr 
wT  =  -  for  section  "ob  " 


r 
w'T2  =         •        for  section  "ran" 


-B)]db 
r 

6dbfs 
If  kw  =  w1,  then  T  =       where  w  =  w1  as  in 

gj  £    kf   figures  A,C  and  D, 

k  =  1  and  T=  - 
f 

Very  often  d  =  2  db,  figures  A,C  and  D,  hence 

3dfs 
T  '  ~       where  T  is  given. 


fg  w 

f[B  +  —  (pb-B)l 

wr 
X  and  Y  *  the  component  reactions  of  the  trunnion 

(See  Chapter  V) 

When  no  rocker  is  used,  the  entire  trunnion  of 
width  "w"  usually  has  bearing  contact  in  the  top 
carriage  trunnion  bearing.   The  design  should  be 
based  on  a  consideration  of  both  the  allowable 
bearing  pressure  f^  and  the  strength  at  section 


973 


"mn"  where  the  trunnion  meets  the  cradle  or  trun- 
nion bracket.   We  have, 

fbw  Dt  =  /X2+Y2  for  bearing 
J   ;.  !-•  pressure 


16  wDt 

mn 

Combining, 

16(X*+Y2) 


X2+Ya   for  strength  at  section 


nf  f, 


Therefore,  assuming  dt,  we  immediately  obtain  Ot 

When,  however,  a  rocker  is  used  the  dimensions 
depend  upon  the  rocker  bearing  length.  Let 

»r  =  length  along  trunnion  for  rocker  bear- 
ing 
wc  =  length  along  trunnion  for  top  carriage 

bearing 
X  and  Y  =  top  carriage  component  bearing 

reactions 

Xr  and  Yr  =  component  rocker  reactions 
a=  distance  from  mn  to  the  center  of  top 

carriage  bearing 
b  =  distance  from  mn  to  the  center  of  rocker 

bearing 
Mx  =  the  bending  moment  at  section  mn  in  the 

plane  of  the  X  component  reactions. 

MV  =  the  bending  moment  at  section  mn  in  the 

plane  of  the  Y  component  reactions. 

wc          wr 

Then  w  =  H-+W-     a  =  w-+  —      b  =  — 

2  2 

Mx=Xa+Xrb     My  =  Ya  +  Yrb  and  M  *  /H2+M« 
then  fb«cDt  =  /  X2+Y*   at  the  top  carriage  bearing 
fbwrOt  =  /  XJ+Yf,  at  the  rocker  bearing 


974 


— W 


TRUMHIOMS 
PLATE      6 


976 

32  M  D* 


a,*  f  ,   ,_    -    Purther  ,  .  SSMA^ 


Since  a  direct  solution  for  D^  is  complicated 
a  trail  solution  is  preferable.   A  reasonable 
procedure  would  be  to  solve  for  Dt  from  the  bend- 
ing equation  assuming  arbitrarily  values  for  Wy 
and  Wr.   Then  knowing  Dt  approximately  we  may 
solve  Wv  and  Wr  in  consideration  of  the  allow- 
able bearing  stress,  and  then  recalculate  Dt. 

TRUHNIOH  BRACKETS: 

In  the  design  of  trunnion  brackets,  we  have  one 
of  two  types: 

(1)  Where  the  bracket  is  secured  to 

a  recuperator  forging,  the  bracket 

merely  transmitting  the  trunnion 
load  to  the  forging,  the  Latter  of 
which  is  stiff  enough  to  carry  the 
bending  stresses.   Plate  VI, fig.  Q. 

(2)  Where  the  bracket  is  secured  to 

light  built  up  cradle,  as  in  Plate 
VI  fig.  B. 

In  the  latter  case  the  bracket  acts  as  a 
stiffener  and  takes  up  the  cross  wise  bending,  the 
longitudinal  shear  reaction  being  transmitted  to 
the  cradle  only. 

For  brackets  of  type  (2),  assuming  the  cradle 
merely  to  take  up  the  shear  reaction  of  the 
rivets  only,  we  find  a  critical  section  "mn"  at  the 
bottom  of  the  bracket,  Fig.  R,  Plate  VI 
Section  "mn"  is  subjected  to: 

(1)  Cross-wise  bending  =  Y  - 

2 
d 

(2)  Longitudinal  bending  -  X  - 

2 


976 


(3)     A  shear  stress  = 

In  brackets  that  are  bolted  to  a  recuperator 
forging  as  in  fig.  Q,  Plate  VI  we  have  grooves  or 
guides,  which  engage  in  corresponding  guides  or 
grooves  in  the  recuperator  forging.   The  pro- 
jections and  grooves  must  be  designed  to  withstand 
the  allowable  total  bearing  stress  and  total  shear 
as  well,  both  of  which  equal  the  X  component  of 
the  trunnion  reaction.   The  bolts  which  secure  the 
brackets  to  the  forging,  merely  take  up  the  tension, 

due  the  moment  caused  by  the  overhang  of  the  trun- 
nions and  the  trunnion  reaction. 

In  the  design  of  trunnion  brackets,  we  have 
other  critical  sections,  as  ab  of  fig.  Q  and  cd 
of  fig.  R,  that  is  just  above  or  at  the  first  row 
of  rivets. 

In  the  design  of  a  trunnion  bracket,  the  critical 
section  is  near  the  first  row  of  rivets, as  sections 
ab,  cd  or  mn  respectively  in  fig.  R. 

The  straining  action  at  either  one  of  these 
sections  consists  of: 

(a)  A  direct  pressure   (or  tension) 
due  to  the  Y  component  of  the 
trunnion  reaction. 

(b)  A  shear  stress  equal  to  the 
X  component. 

(c)  A  bending  moment  in  the 
longitudinal  plane  (due  to  the 
moment  of  the  X  component  =  Xr. 

(d)  A  bending  moment  in  a  cross 
sectional  plane  due  to  the  Y 
component  =  Y  g. 

(e)  A  torsional  or  twisting 
moment  due  to  the  X  component 
*  X  m. 

If  now  Ix  and  iy  are  the  moments  of  inertia  of 
critical  section  and  dx  and  dy  are  the  distances  to 


977 


the  extreme  fibres  in  the  longitudinal  and  cross- 
wise directions,  respectively,  we  have, 

xrdx   Ygdy   Y 
f  =  —  :  —  i  —  :  —  ±  -  for  the  maximum  fibre  stress 

t    —         y 

To  design  for  the  proper  distribution  and  re- 

quired strength  of  rivets,  for  brackets  of  type  2, 
fig.R,  we  assume  a  differential  rotation  about  the 
bottom  of  the  bracket.  Obviously  the  shear  strain 
for  the  upper  rivets  is  a  maximum. 

If  now,  the  vertical  distance  from  the  bottom 
to  the  top  row  is  r0,  for  the  next  lower  row  r,  and 
so  on,  then  for  the  shear  in  the  various  rows  of 
rivets,  we  have,  SQ  =  c  r0,  St  =  c  rt,   Sa  =  c  ra 

etc.     Further  if  we  have  no  rivets  for  the  top 

row,  n  for  the  next  row  and  so  on,  then  taking 
moments  about  the  bottom,  we  have 

c(n0r0+ntr**n8r*  ---  nnr*  )  =  X  r  where  r  is  the 
vertical  distance  from  the  center  line  of  trunnions 

to  the  bottom  of  the  bracket.    Hence 

Xr 
C  =  -    therefore,  assuming  nont  ---  nn 

Doro+ntrt  "  "nnrn   respectively  and  the  spacing 
of  the  rows  r0rt  ---  rn  respectively,  we  obtain  C. 

The  shear  stress  for  any  one  rivet,  becomes, 

X  r  r0 
S  =  ———————  for  the  top  rivets, 

noro+ntrt~  n  rn 


for  the  next  row, 
---  nnrn 


X  r  r 
Sn  =  ; for  the  bottom  row, 


- 
The  tensions  in  the  rivets  or  in  the  bolts  as 

in  fig.  Q,  are  obtained  by  an  exactly  similar  de- 
flection method.  -  (See  design  of  bolts  for  pedestal 


978 


— T* 


FK3.  S 


ELEVATING     ARC 
PLATE     7 


979 


FU5.  W 


FIG      X 


TOP     CARRIAGE 

PLATE  8 


980 


SfCT/OM 


C/l/?/?//lG£ 


PLATE:  9 


981 


lounts  -  External  Forces). 

TOP  CARRIAGES.     The  top  carriage  sustains  the 

tipping  parts  and  during  the  firing 
takes  up  the  reactions  of  tbe 
tipping  parts.  These  loads  are  ap- 
plied at  the  trunnions  and  elevat- 
ing arc  respectively.   With  balancing  gear  introduced 
for  high  angle  firing  guns,  we  have  an  additional 
reaction  due  to  the  balancing  gear.   These  loads 
are  balanced  by  the  supporting  forces  at  the  travers- 
ing pintle  and  at  some  other  contact  with  the  base 
plate  or  bottom  carriage,  the  arrangement  and 
position  of  which  determine  to  a  considerable  degree 
the  type  of  top  carriage  used. 

With  large  caliber  guns,  where  the  design  of 
the  top  carriage  depends  primarily  on  strength  con- 
siderations, special  effort  should  be  made  to  throw 
the  greater  firing  load  on  the  trunnions,  the  ele- 
vating arc  reaction  merely  balancing  the  moment  of 
the  weight  of  the  recoiling  parts  out  of  battery. 
Then,  the  elevating  gear  and  balancing  gear  re- 
actions become  minor  forces  as  compared  with  that 
sustained  at  the  trunnions.   Therefore,  in  a 
preliminary  layout,  we  have  the  load  at  the  trunnions 
balanced  by  the  supporting  reactions  at  tbe  pintle 
bearing  and  clip  bearing  respectively. 

Having  determined  the  external  reaction,  we 
may  examine  critical  sections  and  determine  their 
respective  strengths.   By  classifying  the  various 
types  of  top  carriages  used,  certain  important  sect- 
ions and  the  loadings  on  then  may  be  pointed  oat. 

We  have  tbe  following  types  of  top  carriages:- 

WITH  MOBILE  ARTILLERY. 


(1)     Top  carriages  with  side  frames 
and  connected  together  by  transoms 
or  cross  beams  supporting  the  pintle 
bearing  for  traversing  the  top  carriage, 
(Plate  8,fig.l) 


982 


-  -|~3- 


u 


.   V 


TRAVERSING  GEAR 


PLATE  10 


(2)  Pivot  yoke  type  of  top  carriage, 
the  pintle  bearing  fitting  in  tbe 
wheel  axle  and  prevented  from  over- 
turning by  a  bottom  pin  fitting  in 
an  equalizer  bar  tbe  latter  being 
connected  to  the  trail,   Pivot  yoke 
type  of  top  carriage,  when  used  in 
a  pedestal  mount,  is  prevented  fron 
overturning  by  a  sufficient  shoulder 
at  tbe  top  of  tbe  bearing. 

(3)  Cantilever  top  carriage  used  wben 
balancing  gear  is  introduced.  Tbe 
trunnions  being  at  tbe  rear,  tbe 
pintle  bearing  at  tbe  center  and  front 
clip  bearing  at  tbe  front,  gives  a 
cantilever  loading  on  tbe  top  carriage. 

WITH  TllgP  STATIOHABT  MOUNTS. 

(1)  Pivot  yoke  type  of  top  carriage 
with  small  pedestal  mounts. 

(2)  Side  frame  top  carriages  with 
barbette  mounts. 

WITH  BAILVA7  ABTILLiRY. 

(1)  Pivot  yoke  or  side  frame  top 
carriages  with  pedestal  or  barbette 
mounts,  seated  on  the  car  frame. 

(2)  Side  frame  girders,  supporting 
tbe  trunnions  of  tbe  tipping  parts, 
directly  on  tbe  girders,  and  the 
girders  in  turn  being  supported 

by  tbe  truck  reactions,  or  by  a  dis- 
tributed support  of  special  rail*. 
(1)     Side  Frame  Top  Carriages: 

Side  frame  top  carriages  consist  of  two  side 
frames  either  of  cast  steel  or  of  built  up  structural 
steel.  The  frames  are  connected  together  by  a  beavy 
transom  or  cross  beam  which  contains  tbe  pintle  bear- 
ing for  traversing.   Tbe  pintle  bearing  and  transom 


984 


are  usually  located  either  directly  below  or  to  tbe 
front  of  tbe  trunnions.  Tbe  pintle  bearing  is  de- 
signed only  to  take  up  a  part  of  tbe  vertical  and 
tbe  entire  horizontal  component  of  tbe  reaction  of 
the  tipping  parts,  tbe  overturning  moment  being 
balanced  by  tbe  reactions  of  either  front  or  rear 
circular  clips.   Either  a  rack  or  pinion  gear 
is  introduced  at  a  given  radius  on  tbe  top  carriage 
for  traversing  about  tbe  pintle.   A  pinion  or  worm 
wheel  bearing,  for  tbe  pinion  or  worm  engaging  in 
tbe  elevating  arc  is  located  at  a  given  radius  from 
tbe  trunnion  axis.   In  tbe  design  of  large  guns, 
special  effort  sbould  be  made  to  throw  tbe  greater 
load  on  tbe  trunnions,  tbe  elevating  bearing  merely 
sustaining  tbe  moment  of  tbe  weight  of  tbe  recoil- 
ing parts  out  of  battery. 

External  Reactions;  See  Plate  H  fig.(l) 

As  a  first  approximation,  assuming  tbe  entire 
firing  load  to  be  applied  at  tbe  trunnions,  vie  have, 

2H=K  cos  0 

p  (Ibs)  approx. 

2V=K  sin  0+Wt  I 

where  H  and  V  =  the  horizontal  and  vertical  load 

applied  to  either  trunnion 

0.47  wr 
K  =  tbe  resistance  to  recoil  =  — —  —  VJ   approx 

b   g 

vfv+4700vy 
vf  '        ~  ;   w  =  weight  of  shell 

w  =  weight  of  powder  charge 

w_  -  weight  of  recoiling  parts 

~ 

b  =  length  of  recoil  (ft) 
Wt  =  weight  of  tipping  parts 
Tbe  pintle  reactions,  becomes, 
Ha=2H=K  cos  0  (Ibs) 

2V  lt-2H  bt  lt          ht 

V_  *  -^— — — —  =  (K  sin  0+W*)-r-  -  K  cos0.  r— 

1 

(Ibs) 


85 


SE:CT\ON   A-B 
TYPICAL.   SECTIONS  OF  GIRDERS 


o      o       o 
o       o      o 


o        o         o 
o        o        o\ 


f'.CCCO 


PLATE    II 


986 


and  the  rear  clip  reaction,  becomes, 

2Vb«K  sin0+Ht-Va   (Ibs)  for  the  clip  reaction  on 

either  girder. 

Structural  Steel  Sections  of  Side  Frames: 

Structural  steel  side  frames  are  becoming 
standardized  types  of  side  frames.  We  have  two  types 
of  section,  ( 1)  box  girder  types  and  (2)  simple  neb 
and  flange  section  .   The  advantage  of  a  box  girder 
type  is  that  stiffeners  are  not  needed  and  only  one 
cross  beam  or  transom  is  required  the  frames  being 
sufficiently  rigid.   The  fabrication  of  such  however 
is  more  complicated,  than  with  simple  web  and  flange 
sections. 

After  a  layout  contour  is  made  of  the  frame 
several  sections  should  be  taken  as  (m-n)  Platell, 
fig.U). 

A  typical  box  girder  shown  in  Platell,  fig.  (2) 
and  a  typical  flanged  web  section  is  shown  in  Plate 
11  fig.  (3).   As  a  first  approximation  it  will  be 
assumed  that  the  flanges  take  all  the  bending  stress 
and  the  webs  the  entire  shear.   If 

I  =  moment  of  inertia  of  section 

d  3  depth  of  flange  (in) 

A  =  area  of  flange  (sq.in) 

fm  =  max.  allowable  fibre  stress 


(Ibs/sq.in) 


tbeD          d«        Ad» 

2A  —  =   

-   I        ap 

4            w 

d 

My               Vj 

b    •'•mn  2 

I 

v 

1 

T 

b    1ran 

A  d 

1                        b     mn 

-  (.sq.in) 

(in) 


Thus  with  a  constant  flange  section,  we  must  in- 
crease the  depth  of  the  girder  as  the  distance  from 


987 


the  reaction  V^. 

If  on  the  other  hand  for  construction  considerations 
and  approximately  constant  depth  of  girder  is  re- 
quired, the  flange  area  must  be  increased  with  the 
distance  lmn.   These  factors  determine  the  cross 
section  of  girder,  the  area  of  the  flange  and  the 
depth  of  girder.   The  depth  of  girder  should  "be 
sufficient  with  a  given  thickness  of  web  "t",  not 
to  exceed  the  maximum  allowable  shear  stress  fs. 
Since  the  shear  on  the  web  is  practically  uniform, 
we  have,  fs  dt  =  V^  for  one  web  and  2fg  dt  =  V^ 
for  two  webs  as  in  a  box  section,  then 


d  =  one  web,   d  *  •     box  section 

fat  2fgt 

Pitch  of  Rivets  in  built  up  girder: 

Let  p  =  pitch  of  rivets  (in) 

R  =  allowable  total  shearing  stress  on  rivat 

d  =  depth  of  web 

F  »  total  shearing  force 
Then  considering  a  portion  of  the  web  of  length  p 

of  a  compound  I  section,  we  have,  Fp  *  Rd,  where  P=Vb 

p  j 

or  the  total  shear  on  the  section,  hence  p  »  —  (in) 

r 

Now  for  one  web  and  two  angle  irons  connecting  the 
flange  plates  with  web,  we  have 

n         ft 

R  =  2  -  d*  fsr  *  \\  dv  fsr  where  dr  =»  diaro.  of  rivet 

(in) 

fgr  *  allowable  shear 
stress  on  rivet 
(Ibs/sq.in) 
With  a  built  up  box  section  as  in  Plate  11  fig. 4, 

R  -  2  J  d«  fir  -  \  d-  fsr 

fig. (5)  R*4-  d*  fsr  =  n  d£  fsr 

4 


988 


GENERAL  DESIGN  PROCEDURE. 


Type  of  gun 
"Howitzer  or  Can" 


=  155  M/M  Howitzer 


Type  of  Mount 
"Field  Carriage" 
"Platform  Mount" 
"Caterpillar  Mount" 


=  Field  Carriage 


Diam.  of  bore 
d  (inches) 


=  6.1 


Muzzle  Velocity 
v  (ft/sec) 


1850 


Weight  of  Projectile 
w  (Ibs) 


=  95 


Weight  of  ponder  charge 
w  (Its) 


14.25 


Weight  of  Recoiling  parts 

•f 


=  4200 


Max.  pressure  on  breech 
Pbn  (Ibs/sq.in) 


=  30000 


Length  of  Recoil  Max. 
Elevation  bs  (ft) 


Max.  angle  of  elevation 
0B  (degrees) 


65'< 


Assumed  length  of 
horizontal  recoil 


(ft) 


=   4 


S8S 


Min.  angle  of  elevation 

0i  (degrees)  --50 

Travel   of  projectile  up  bore 

u    (inches)  s   117.5 

Area  of  bore 

A   =  0.786  d*    (sq.in)  =  29.75 

INTERIOR  BALLISTICS. 

Area  of  bore  of  gun 
A  =  0.785  d'(sq.in) 

Total  max.  pressure  on 
breech  of  gun 


Mean  constant  pressure  on 
projectile 

*v'*12  n.  , 
Po  =  -rr:  —  dfes) 

644u 

Tine  Abscissa  of  max. 
pressure 

...c  (2^s-i) 

16  pe 


L-  ^-  -£=)-U  (in) 
16  Pe 


Pressure  on  base  of  breech 
when  shot  leaves  muzzle: 

(Ibs) 


Max. Velocity  of  free  recoil: 

wv+4700  ii  ..4  ,   x 
Y  =  (ft/sec) 


990 


Velocity  of  free  recoil  when 
shot  leaves  muzzle: 

(w+0.5  w)v 
V£o  = (ft/sec) 


Time  of  travel  of  shot  to 

muzzle: 

a.   u 


Time  of  free  expansion  of 
gases  : 


OD 


32-2 


Free  movement  of  gun  while 
shot  travels  to  muzzle 

<»> 


Free  movement  of  gun 
during  powder  expansion 

pob   fo 
xf'o=  T~  g  "7"+vfotin  (ft) 


Total  free  movement  of  gun 
during  powder  pressure  period 

E*Xfo+Xf,0  (ft) 


Total  time  of  powder  pressure 

period 

T  =  t  »t 


991 


STABILITY:    TOTAL  RESISTANCE  TO  BICOIL  AT 


MAXIMUM  AND  MINIMUM  ELEVATION, 


Weight  of  system  (gun  and 
carriage)  Ws  (Ibs) 


Distance  from  spade  point 
to  line  of  action  of  YJS 
(from  preliminary  layout) 


Height  of  trunnion  from 
ground  (assume)  ht  (ft) 


Horizontal  distance  from 
spade  point  to  trunnion 
center  (assume)  lt  (ft) 


Distance  from  center  of 
gravity  of  recoiling 
parts  to  trunnion 
(assume)   s  (ft) 


Moment  arm  of  resistance 
to  recoil  for  angle  of 
elevation  0 
d  =  bt  cos0+s-ltsin0  (ft) 


Height  to  center  of  gravity 
of  recoiling  parts  for 
horizontal  recoil 
b  -  h+s  (ft) 


APPHOIIMATI  CALCULATION; (g  and  T  not 

comp  u  t  e  d) 

Velocity  of  free  recoil 

wv +47005 

Vf  -  

I  WJ 


992 


Travel  up  bore   u  (inches) 


Initial  recoil  constrained 
energy  (approx) 

Ar  =  -  —  V*  (ft/lbs) 

2  8 

where  Vr  =  0.92  Vf  (approx) 

long  recoil 
=  0.88  Vf  short  recoil 


Displacement  of  gun  during 
powder  period 

.w+0.5w.  u   ,   . 
E_  =  ( )  —  (ft) 

wr    12 

where  a  =  2.25  for  long  recoil 
=  2.22  for  short  recoil 

(1)     Constant  resistance  throughout 
Recoil . 

Constant  of  horizontal 
stability 

Overturning  moment 
Stabilizing  moment 

(Usually  assume  0.85) 

Kin.  length  of  recoil  con- 
sistent with  stability  at 
minimum  elevation 


U1U   p*9V«.b*V/U        i 

Wsls+WrErcos0-/(Nsls+WrErcos0)2-4Wrcos0(WslsEr- 


c 


2   YTr   cos   0 


At  0°  Elev.  cos  0=1  and  d  =  h 


(ft) 


Max.  alienable  recoil  at 
horizontal  elevation 


.035  Vf/b  (ft) 


Assumed  length  of  horizontal 
recoil  at  min.  elevation 
bb   (ft) 


Total  resistance  to  recoil  at 
horizontal  or  minimum 


use  Ar  for  long  recoil 


Assumed  length  of  recoil  at 
nax.  elevation  consistent 
Hith  clearance  bs  (ft) 


Total  resistance  to  recoil 


at  nax.  elevation  (0ffi  = 


Use  Ap  for  short  recoil 


993 


Variable  Resistance  to  recoil 


Constant  of  horizontal 
stability  Cs 


Min.  length  of  recoil  con- 
sistent with  stability  at 
min.  elevation 
1 

b 


Bf.)]   (ft)  at  0°  elev.  cost=l  and  d=h 


994 


Max.  allowable  recoil  at 
horizontal  elevation 

bh   -.035  Vf  /n~  (ft) 
"max 


Assumed  length  of  recoil 
at  horizontal  or  min. 
elevation   bh  (ft) 


Mean  resistance  to  recoil 
during  retardation  period 


Stability  slope 
Wrcos0 

•JHI 

d 


Cs  • (Ibs/ft) 


Mean  resistance  to  recoil 
in  battery 

K  -KB+  ~(b-Er)  (Ibs) 


Mean  resistance  to  recoil 
out  of  battery 

k  -KB-  5(b-Er)  (Ibs) 
2 


Exact  calculation  E  and  T  computed  (See  Interior 
Ballistics) . 

(1)     Constant  resistance  to  recoil. 

Constant  of  stability 

(assumed)  Cs  •  = 

(C-»0.85  usually) 

9 

A  =W  cos  0mi  = 


996 
B*Wrcos«fmin(VfT-E)-Wsls 


C  *    Wsls(VfTHS)+   i  -£  V  f 


cc 


Min.  length  of  recoil  con- 
sistent with  stability  at 
mm.  elevation 

-B+/B*-4AC  /f 
— (ft) 


Allowable  recoil  at  horizontal 
elevation 

bh    =.035  /h~  (ft) 
nmin. 


Assumed  length  of  recoil 
at  minimum  elevation 
bh  (ft) 


Total  resistance  to  recoil 
at  min.  elevation 


Kh 


Max.  elevation  consistent 
with  clearance  bs  (ft) 


Total  resistance  to  recoil  at 
max.  elevation 


996 

(2)     Variable  resistance  to  recoil 
Constant  of  stability  (assumed)  C_   = 


WPcos*5 
Stability  slope  ffl=C  — (Ibs/ft) 

S   d 


Total  resistance  to  recoil  during 
powder  period  consistent  with 

stability 

C8(*8l.-WrE  cos  0) 


(lbs) 


COS 


A  =  m 


B  ,  •£!!  -2K-2BE 


K*mT8 

C  =(2E 2  VfT)K  +  — ~- 

Br  4ro* 


Min.  length  of  recoil 
consistent  with  stability 
at  min.  elevation 

•  .  =Bij£i*L  (f t ) 


Allowable  recoil  at  horizontal 
elevation 

bh    =  .035  Vf  /~h   (ft) 
"max-. 


997 


Assumed  length  of  recoil  at 
minimum  elevation   bh  (ft) 


Total  resistance  to  recoil  during 
powder  period  with  assumed  length 
of  recoil  at  min.  elevation 

mrVf+ai(b-E)2 
Kh 

2fbh-E+VfT-  -  —  (b-E)] 
2  ro 


Total  resistance  to  recoil  in  out 
of  battery  position  with  assumed 
length  of  recoil  at  min.  elev. 


Margin  of  stability  at  minimum 
elevation  for  the  assumed  long 
recoil  in  and  out  of  battery 
respectively. 

Mean  constant  pressure  on  breech 
of  gun 


'be 


1.12 


Max.  overturning  force  in  battery 
(stability  limit) 


l-2ndr 

n  =  0.15  to  0.25 

1  =  k  ,  3  clips  =  b,  4  clips 

Ci 

dr  =  mean  distance  to  guide  friction 
from  bore. 


998 


Max.  overturning  force  out  of 
battery 

*«l«~Wf  bh  cos  0 
k{  «  -2-2 — F. — !] 


Margin  of  stability  in  battery 
K-R   (Ibs) 


Margin  of  stability  out  of 
battery  k'=k  (Ibs) 


Estimated  Jump  of  Carriage  at  Horizontal  elevation, 

Distance  from  spade  to  center 

of  gravity  of  Wg  —  ds  (ft)      = 

Time  of  recoil  (approx.) 

"r  vf 
t  «  — -  — •  (sec)  = 


Ang.  vel.  about  spade  at 
end  of  tine 


t 
g(Knd-Wgls)ti 


(rad/sec) 


Tine  to  nax  .  lift  of  carriage 
from  end  of  time  t 
d| 


t  *  -  w   (sec) 


Total  angular  displacement  about 

spade  to  max.  lift. 

i 

6  =  -  w  (t  +t  )  (rad) 


999 


Lift  of  wheel  from  ground 
Sw  =  Iwe=l8  6  (appro*)  (ft) 
where  lw  =  distance  from  spade  to 
wheel  base  (ft) 


Potential  energy  at  nax.  lift 
E8=WS18  6  (ft/lbs) 


VABYIKe  TBK  RECOIL  OB  BLKVATIOMt 

In  general  assume  the  length  of  recoil  at 

horizontal  recoil  constant  from  £)j  to  0}  degrees, 
(usually  from  0°  to  20°  elevation),  then,  decrease 
the  recoil  proportionally  with  the  elevation,  or 
consistent  with  clearance. 


Length  of  intermediate  recoil  (ft)  from  A^  to  0m 

b  b 
b  =  -  (£H0n)+bs 


degrees.      b  b 


ri  i 

bh  =  long  recoil 
b8  =  short  recoil 

Resistance  to  recoil:  from  0j  to  0j 

Variable  Kn  =  

2[bh-E+VfT-  -  ~(b-E)] 

o    - 

Lj         ID  ft 

] 

Constant  Kh  **  — — — — 
b-E+VfT 

from  £5  2  to  ^ 

assumed  constant  =  K  =  — -^——    exact 

b-E+VfT 


1000 


0.47 


Moment  arm  of  resistance  to  recoil  about 
spade   d  =  htcos0+S-ltsin#  (ft) 


«  •»* 

• 

a     • 

a. 

> 

C.' 

0      -H 
i-l      W 

0       • 

4*    M 

o   *• 

V)       41 

n   10 

I/I 

H 

M 

p>  1i 

•H 

in  i-i 

6  o  «• 

ll    +>     P    -O 

4*        0 

v    w 

Q)     *rt 

(V            0 

0 

•»i     h 

h    O 

•  .a 

.        _ 

rH        • 

o 

•*»*»* 

9  "& 

•H        0         6 

•H     *J      • 

rH     4) 

a   i/i        «> 

ft 
00 

a 

00 

a 
o 

•a 

JB        -H 

<0       * 
•»»      * 
CO      > 

4>      » 

(0    C 

4»      0 
(O    O 

(0     *• 

V 

o  o 

O     -rl    H    « 

E      «0    •*!      » 

0     «    0     P. 

»-*»•• 

0. 

bh 

CsWrcos0i 

0 

Kh 

«i 

Usually 

J 

* 

from  0° 

• 

# 

. 

• 

. 

* 

to  20° 

jj 

bh 

CsWrCCS01 

0 

K 

Q     • 

di 

01 

bh 

0 

0 

Kh 

^ 

From  20° 

to  max. 

• 

• 

• 

• 

• 

• 

elev  . 

0 

b 

0 

0 

K 

d 

L 

i. 

0 

0 

«* 

* 

1001 


Initial  recuperator  reaction 
(appro*. )  (Max.  elev.  =0m) 

Fvi=  1.3  Wf(sin  Om+0.3cose(in)(lbs) 


5623.8 


Min.  Bean  recuperator  reaction 
(Max.  elev.  0m) 

FVB)=2Wrrsin0m-0.3(l-cos0m)+0.3Wrl 

(Ibs) 


7418 


Min.  allowable  ratio  of  com- 
pression 

P^  _   1.5(1.665Fvm-Fvi) 

r,,^          fm* 


M 


1.79 


Max.  allowable  ratio  of  com- 
pression (stability  limit) 


"max 


VI 


VI 


=   9. 


Max.  allowable  ratio  of  com- 
pression (heating  limit) 
m  =  2  to  2.5 


Mean  temperature  in  recuperator 

(assumed) 

!„,  =  20°  to  30°  C  (centigrade) 


Max. temperature  due  to  com- 
pression 

where  k  =  1.3  with  floating  piston 
=  1.1  air  contact  with  oil 


1002 

Ratio  of  compression  used. 
M 


Max.  allowable  air  pressure 
Pafa  =  200°  to  250° 


Final  air  pressure 
paf   (Ibs/sq.in) 


Initial  air  oressure 

Paf 
Pai  =  "      (Ibs/sq.in) 

ID 


Initial  recuperator  air 

e 

P, 


volume  required: 


vi    ,  m     . 
V_  =  bj,(— - — — • )   (cu.in) 

Pai     i 


m 


k  - 


bn  *  length  of  horizontal  recoil 
(inches ) 


Effective  area  of  recuperator 
piston 

Fvi 

A.,  =  (Ibs/sq.in) 

Pai 


Length  of  air  column  in  terns 
of  recoil  stroke 


j  =  0.8  to  1.2  usually 


Actual  length  of  air  column 
1  * '  j  bn  (in) 


1003 

Air  cyl.  cross  section       Aa 

Ratio  of  -  =  — 

Effect.  area  of  recuperator   Av 

Aa   1,  m* 

r  =  —  =  -(  —  •  -  )  =2.8 

Av    j   J 

•   -  1 

Area  of  cross  section  of  air 
cylinder 

Aa  =  r  Av  (sq.in)  = 

Max.  allowable  fibre  stress  in 
the  recuperator  piston  rod 

f_  (Ibs/sq.in)  = 

a     t 
fD  =  -  to  -  elastic  limit  usually 

Required  area  of  cross  section  of 
recuperator  piston  rod 


=  1.2  -~  (sq.in) 


Required  diam.of  recuperator 
piston  rod 

(in) 
0.7864 


Assumed  diam.  of  recuperator 
piston  rod 
dv  (in) 


Area  of  cross  section  of  re- 
cuperator piston  rod 
a  =  0.7864  d 


Required  area  of  recuperator 

cylinder 

A    -A  +av  (sq.in) 


1004 


Required  diao.  of  recuperator 
cylinder 


vo 


'vo 
0.785 


(in) 


Assumed  diam.  of  recuperator 
cylinder  dvo  (in. ) 


Area  of  recuperator  cylinder 
A    (sq.in) 


Effective  area  of  recuperator 

piston 

Av  =  Avo-av  (sq.in) 


Initial  recuperator  pressure 

P. 


ai 


Fvi 

=  (Ibs/sq.in) 

A 
v 


Final    recuperator  pressure 
Paf   =   m  Pai      (Ibs/sq.in) 


Initial    air   volume    (exact) 
i 


''"  ' 

k=1.3  to  1.1 


• 


Length   of   air   col  uran(exact ) 
1    -  -      (in) 


1006 


RECOIL  BRAKE  LAYOUT. 


Max  .hydraulic  pull  (at  max.  elev.) 

-F    (Ibs) 


Max.  hydraulic  pull(at  0°  elev.) 
Phc=Kh-0.3Wr-Fvi  (Ibs) 


Max.  allowable  brake  pressure 

ph  max.  (Ibs/sq  .in) 

Ph  max  =  400°  to  500°  (lbs/sq.  in) 


Required  effective  area  of  recoil  piston 

Phm 
A  =  -  (sq.in) 

Ph  max 


Reciprocal  of  contraction  factor  of 
orifice  assumed   C 

Win.  recoil  throttling  area 


C  A  Vr 


Bin 


max 
where  Vr  =  0.9  Vf  (approx.) 


Hydraulic  brake  pressure  (at  0°  elev.) 

pbo 
pho  =  -7-  (Ibs/sq.in) 


Max.  recoil  throttling  area  (at  0°  elev.) 

C  A  Vr 
Wh  max  =  -   where  Vr  =  0.92  Vf  (approx.) 


The  battery  stabilizing  moment  of  counter  recoil 
WsL,j  =  150  to  250  La  (inch  Ibs.) 


1006 


L£  *  distance  from  wheel  base  to  Ws 

L&  -  distance  from  spade  to  wheel  contact 

Max.  buffer  reaction  of  counter  recoil 
"SLB 


DIMIB8IOR8  07  HOLLOW  PISTON  BODS. 

Max.  allowable  buffer  pressure 

pb'm  (Ibs/sq.in) 

Assume  from  1600  to  2500  (Ibs/sq.in) 

Area  of  buffer  chamber 

A.  =  JLL   (sq.in)  « 

b 


Required   inside   diam.of   piston  rod 

/     *b 
0.7834 

Area  of  inside  cross  section  of  rod 
Filloux  recoil  system 

Abs  3  wbn  (sq.in) 

Required  inside  diam.  of  piston  rod 
Filloux  recoil  system 


0.7834 


Assumed  inside  dian.  of  piston  rod 
d   (in) 


1007 


Max.  allowable  fibre  stress  brake  piston 
rods   fm  (Ibs/sq.in1) 

^  to  J  elastic  limit  (Ibs/sq.in) 

Outside  diam.  of  piston  rod  based  on 
max.  allowable  tension 


«  /df  +1.6  —  (in) 


Outside  diam.  of  piston  rod  based 
on  max.  hoop  compression 


°  .. 


Assumed  outside  diam.  of  rod  do  (in) 

Area  of  total  cross  section  of  rod 
ar  •  0.7854  d«  (sq.in)  - 

(2)     Dimensions  of  Solid  piston  rods 

Max.  allowable  fibre  stress 
fB  (Ibs/sq.in) 

3      1 

-  to  j  elastic  limit  (Ibs/sq.in) 

Required  area  of  piston  rods 

pbm 
al  «  1.3    ••  (sq.in)  « 

r. 

Corresponding  diam.  rod 


1008 


Assumed  dism.  of  rod 
dr  (in) 


Area  of  rod  ar  =  0.7854  dj 
(sq.in) 


AREA  OF  DIAM.  OF  RECOIL  OYL1BDIB 


Required  area  of  recoil  cylinder 
AJ=A'+ar  (sq.in) 


Corresponding  diam.  of  recoil  cylinder 

is 


Assumed  diam.  of  recoil  cylinder 
d  (in) 


Area  of  recoil  cylinder 
A.  =  0.7864  d*  (sq.in) 


Effective  area  of  recoil  piston 
A  =  Ar-ar  (sq.in)          = 


Max.  pressure  in  recoil  cylinder 


Phm 


max 


PRINCIPLE  RBACTION8  AND  STRB8SEE  THROUGH- 


MOUNT. 


Total  resistance  to  recoil  at 
max.  elevation 

imrvr 
K,  =          (Ibs) 


1009 


Initial  recuperator  reaction 
Fvi=1.3Wr(sin0m+0.3cos0m)   (Ibs)   = 

Max.  hydraulic  pull  (approx.) 
max.  elev. 


FROM  RECUPERATOR  AND  RECOIL  BRAKE  LAYOUT 


DETERKINE: 


Distance  from  axis  of  bore  to  line  of  action  of  P1 

n 


dh  (in) 


Distance    from  axis   of   bore   to   line   of    action   of 


dv    (in) 


Distance  from  axis  of  bore  to  line  of  action 
of  resultant  braking  B 


Distance  between  guide  friction  1  (in)  = 


1  =  —  for  3  clips  (in) 
2 


1  =  bj,  for  4  clips  (in) 

Coordinates  along  bore  of  front  and  rear  clip 
reactions  with  respect  to  center  of  gravity  of 

recoiling  parts 
x±  (in) 
xt  (in) 


1010 


Distance  from  axil  of  bore  to  line  of  action 
of  mean  guide  friction  (from  layout)  dr  (in) 

Total  braking  at  max.  elevation 

Ut-x  1 
K.+Wp(sin0-ncosfl      * 


1 

(Ibg) 


I 
where  n  *  0.1  to  0.2 


Recuperator  piston  friction 

Fvi 
Rpv  «  -04  n  Dvo  T^-  wp 

Assume  W.  •  width  of  packing  (in) 


Recuperator  stuffing  box  friction 

Fvi 
R   -  .04  «  dy  — -  W8   (Ibs) 

Ay 

Assume  wg  «  width  of   packing    (in) 


Total  recuperator  packing  friction 
R(e*p)v(lb«> 

Hydraulic  piston  friction 

K. 

Rph  -  .04  *  D  -jp  irp  (Its) 


Hydraulic  stuffing  box  friction 
Rib  -  ,04n  dp  —  w,  (Iba) 


Total  hydraulic  packing  friction 


1011 
Total  hydraulic  pull  (max.  elev.) 


Total  hydraulic  reaction 
(max.  elcv.) 


Max.  hydraulic  pressure 

phm 
phm  -  (Ibs/sq.in) 

A 


Max.  recuperator  reaction 

Fvf»m  Pvi  (Ibs) 

where  m  *  ratio  of  compression 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


-?5  6     1361 

FfcBi    RECD 


Form  L9-25wi-8,'46  (9852) 444 


THE  MBRARY 


TIP 

6Uo       TJ.S. 


dept.  - 


IJbrary         Sign     Of 

systems  »  •  • 


— 

UC  SOUTHERN  REGIONAL  LIBRARY  FACILITY 


UF 


Ubrary 


ST'     » 

SEP      '73 


